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# Introduction to Ordinary Differential Equations MATH 512

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Introduction to Ordinary Differential Equations Todd Kapitula Department of Mathematics and Statistics University of New Mexico September 28 2006 E mail kapitula mathunmedu Todd Kapitula O H to CA3 g 01 CONTENTS Introduction 01 Notation and introductory de nitions 02 Solving linear systems 03 The phase plane for linear systems 031 032 Complex eigenvalues Real eigenvalues 033 Classi cation of the critical point 04 The phase plane for conservative nonlinear systems Existence and uniqueness 11 Existence 111 112 Proof by polygonal approximations Proof by successive approximations 12 Uniqueness 13 Continuity With respect to initial data 14 Extensibility 15 Examples Linear systems 21 General results 22 Equations With constant coef cients 221 222 223 224 225 226 23 Equations With periodic coef cients 231 232 Example the forced linear Schrodinger equation The fundamental matrix solution The Jordan canonical form Estimates on solutions Linear perturbations stable Linear perturbations unstable Nonlinear perturbations Example periodic forcing 233 Example linear Hamiltonian systems 234 Example Hill7s equation The manifold theorems Stability analysis the direct method 41 The wlimit set 42 Lyapunov functions 421 Example Hamiltonian systems Periodic Solutions 51 Nonexistence Bendixsonls criterion 52 Existence Poincar Bendixson Theorem 521 Examples 53 Index theory 54 Periodic vector elds ODE Class Notes 2 6 Applications of center manifold theory 63 61 Reduction to scalar systems 63 611 Example singular perturbations 66 612 Example hyperbolic conservation laws 66 62 Reduction to planar systems 69 621 The Hopf bifurcation 69 622 The TakensBogdanov bifurcation 70 References 7 0 0 INTRODUCTION Some physically andor biologically interesting mathematical models are a The GrossPitaevski equation 0 21 7 3 vlttgt 139 WM 0 dt2 7 m1an 7 is a model used in the study of BoseEinstein condensates see 9 10 13 and the references therein Here represents the wavefunction of the condensate and represents the applied external potential The 77boundary condition77 guarantees that the condensate is localized ie experimentally realizable b Lotka Volterra competition model dN dN Til 7 T1N1lt17 N1 K1 7 blNlNg T 7 T2N217 NQKQ 7 bQNlNQ Here X represents the carrying capacity of the environment for species Ni in the absence of competition and bi re ects the competition between the two species c Fire y s ashing rhythm d9 d g wAsina7t9 CT 9 Here 9 represents the phase of the re y s rhythm A is the re y s ability to modify its frequency and a is the periodic stimulus Different questions can be asked for each model For example when considering the Lotka Volterra model one can as a does one species wipe out the other b if not in which manner do the two species coexist relatively constant populations or populations which periodically uctuate The purpose of this course is to acquire and develop the mathematical tools that will allow you to begin to analyze models such as the above In the remainder of this section we will quickly review the material that you should have learned in your undergraduate course in Ordinary Differential Equations eg see 1 as well as your introductory course in real analysis eg see 01 Notation and introductory definitions De nition 01 A norm l l R gt gt R satis es a in HA S W W 3 Todd Kapitula b lcml lcl for all c E R c 2 0 and equality occurs only if m 0 De nition 02 Let x 11 i i znT E R A norm is given by n W 1 eril i1 Let A E Rnxn The norm of A is given by lAl I suplAml l Z lam ij1 Remark 03 One has that a More generally a norm can be de ned by n lmlpiZlIilp1p i1 It can be shown that each of these norms are equivalent ie given a 1 S pq S 00 one has that there are positive constants cl and Cg such that Cllmlq S lmlp S CZlmlq For example W2 S lmll S x lmlz Thus the choice of the norm is not important 0 MM S W lmlA De nition 04 Given an mo 6 R and 7 gt 0 de ne BMW 2 m e R l9 7 mol lt v MW m e R l2 7 mol v Regarding calculus for vectors we Write a fmtdt letdtiiifzntdtT b dmdt d11CltClInCltT De nition 05 Let G C R X R be open and let f G gt gt R be continuously differentiable The matrix Df I ifquot1 E Rnxn satis es 6i Dfij 7 f 7 8117 De nition 06 Let f G gt gt R be continuous An ordinary differential equation ODE is of the form d t I i a f 72 d The function x solves the ODE on an open interval I C R if 45 z I gt gt R is continuously differentiable with lt15 m ab ODE Class Notes 4 Remark 07 Consider the secondorder scalar equation 2 7 y y 7 y 7 s1n ti Upon setting 11 I y712 I y39 one gets the rstorder system i1 127 i2 7I1z sint7 53 ft77 11 7 I2 1 7 lt 12 gt7 ft m 7 lt 711z sintgt This trick can be used to transform a scalar equation of order n to a rstorder system with n equationsl where 02 Solving linear systems Now let us refresh our memories as to how one can explicitly solve linear ODEs of the form it Am 01 where A E Rnxnl Substituting 1 Z en 02 into equation 01 yields A 7 A11v 0 In the above the vector v is known as the eigenvector7 and the corresponding eigenvalue A is found by solving the characteristic equation detA 7 All 0 If A E R then the solution with realvalued components is given in equation 02 If A E C ilel7 A a il7 then the corresponding eigenvector is given by v eriq7 where I q E R and the two linearly independent solutions with realvalued components are given by 1 e quot cosbtp 7 sinbt q mg e quot sinbtp cosbt q i If the eigenvalues are simple7 then one can nd n linearly independent solutions ml l l l 7 mn via the manner proscribed above7 and the general solution is then given by it 6111Cnm where Cj E R forj Lianl 03 The phase plane for linear systems Now suppose that n 2 The eigenvalues are zeros of the characteristic equation A2 7 traceAA detA 07 A Ai I lttraceA i xtraceA2 7 4detAgt l 5 Todd Kapitula 031 Real eigenvalues First suppose that traceA2 gt 4detA so that A lt A E El When graphing trajectories we will use the fact that the line in the zy plane through the origin parallel to the vector vltgt d y7zi c is given by If detA lt 0 then A lt 0 lt A and the critical point x 0 is known as an unstable saddle point For example suppose that A lt 3 5 gt i 03 72 74 The eigenvalues and associated eigenvectors are given by 71 75 A 727v1lt1gtiA177 2lt2gt7 so that the general solution is given by 1t cle 2 lt 1 gtC2e lt 3 When Cg 0 the solutions are restricted to the line y 71 furthermore any solution on this line goes to the origin exponentially fasti When cl 0 the solutions are restricted to the line y 72 51 furthermore any solution on this line goes grows large exponentially fasti Sample trajectories are given below Figure 1 The phase portrait for equation 03 If detA gt 0 then signi signtraceA which implies that if traceA lt 0 then m 0 is a stable nodei For example suppose that Alt73 5gt 04 476 The eigenvalues and associated eigenvectors are given by 7 27v1lt71gti 17v2lt7igt7 ODE Class Notes 6 so that the general solution is given by 71 7 75 1tcle Qtlt 1gt02etlt 4gtl When Cg 0 the solutions are restricted to the line y 71 furthermore any solution on this line goes to the origin exponentially fastl When cl 0 the solutions are restricted to the line y 7451 furthermore any solution on this line also goes to the origin exponentially fastl Sample trajectories are given be ow Figure 2 The phase portrait for equation 04 On the other hand if traceA gt 0 then m 0 is an unstable nodel For example suppose that Altj g 05gt so that the general solution is given by 1t cle 2 lt i gtC2equotlt i When Cg 0 the solutions are restricted to the line y z furthermore any solution on this line grows large exponentially fastl When cl 0 the solutions are restricted to the line y 45 I furthermore any solution on this line also grows large exponentially fastl Sample trajectories are given below 7 Todd Kapitula M Figure 3 The phase portrait for equation 05 032 Complex eigenvalues lf traceA2 lt 4detA iiei AiaiiB C BERT then it turns out to be the case that through a clever change of variables the system in equation 01 is equivalent to 9311 Blt 06 We will now study equation 06 in coordinates iiei Y ll 091 i 592 9392 591 0W2 Upon using polar coordinates iiei setting yl Tcost yg Tsint9i and using implicit differentiation y39l i COSO 7 T9sint9 y39g rsint9 T cost it is seen that t y39l 3086 y39g sint9 n9 iy39l sint9 y39g cost9i Simplifying the above yields iiei Tt T0em 9t t 90 Thus the solution to equation 06 is given by wt Tlt0gtem coswt 60 mt rltogtem sinwt 60 where rltogt lm 920 tan6lt0gt W 910 The form of the solution guarantees that the trajectories will spiral about the origin If a gt 0 ie traceA gt 0 then the solutions will spiral away from the origin exponentially fast in this case the origin is an unstable spiral point A sample trajectory is given below ODE Class Notes 8 Figure 4 The origin is an unstable spiral point If a lt 07 ilel7 traceA lt 07 the solutions will spiral towards the origin exponentially fast in this case7 the origin is a stable spiral point A sample trajectory is given below Figure 5 The origin is a stable spiral point Finally7 if a 07 irel7 traceA 07 the trajectories will be closed in this case7 the origin is a centerr A sample trajectory is given below Figure 6 The origin is a center 9 Todd Kapitula The last item to be determined is the direction of spiralling is it counterclockwise or clockwise The answer of course depends on the particular problemi One strategy which will help in answering the question is illustrated in the following example Suppose that A lt i j gt i The eigenvalues are Ai 71 i 2i so that the origin is a stable spiral pointi Write the system as a 1 7 4y 07 y z 7 yr On the halfline z 0 and y gt 0 the vector eld satis es 139 74y lt 0 As a consequence zt is decreasing whenever the trajectory hits this line which implies that the motion is counterclockwise A sample trajectory is given be owl Figure 7 The phase portrait for equation 07 Remark 08 If 139 gt 0 on the halfline z 0 and y gt 0 then the motion will be clockwisei 033 Classification of the critical point As it can be seen in the examples the stability of the critical point depends upon the sign of the eigenvalues The following table summarizes the above discussion Eigenvalues Type of Critical Point A1 A2 gt 0 Unstable node A1A2 lt 0 Stable node A1 lt 0 lt A2 Unstable saddle point A a i i a E Rl Unstable spiral point A a i i a E R Stable spiral point A ii Linear center 04 The phase plane for conservative nonlinear systems Finally let us brie y discuss phase portraits for planar vector elds ie the graphical representation of solution curves to r f9 m e R 0 8 ODE Class Notes 10 For equation 08 suppose that E R2 gt gt R is a rst integral ilel along trajectories d EVEm170l One then has that Emo for all t E R hence solutions reside on level curves and the set of all level curves gives all of the trajectories For example if 1 127911T7 Which arises When equation 08 is equivalent to the secondorder problem a39c39 91 07 then 1 E1 98 ds 0 is a rst integral Here the functional E represents the total energy for the conservative physical systeml When considering the pendulum equation ilel gz sin I one has that l 11 0 sinsdsl In this case if lt 2 then the solution is periodic 11 Todd Kapitula 1 EXISTENCE AND UNIQUENESS Consider the initial value problem lVP 51 ft7917 93t0 9307 11 where f is continuous on an open set C with t0mo 6 G1 When considering the nonautonomous equa tion 111 one can make it autonomous by rewriting it as dm dt 3 937 E 1 07t0 1077 9911 y0yo y917t7 9yft791711 Hence without loss of generality one can consider the IV 93 H91 910 9107 12 where f G C R gt gt R is continuous From this point forward it will be assumed that the ODE under consideration is autonomous and given by equation 12 11 Existence Theorem 11 Peano7s Existence Theorem Let mo 6 G be given There is a 6 gt 0 and a function mt de ned on I 766 which solves equation 12 Remark 12 The solution may not be unique for example consider 139 izi 10 0 which has the family of solutions zt zat where a E R and tlta t 07 0S I I W402 agti 111 Proof by successive approximations Herein an additional assumption on f will be made precisely that f is uniformly Lipschitz continuous on C see De nition 1114 This is for technical reasons only and as it is seen in Section 1112 it can be removed with a different method of proof of Theorem 11 Let 7 gt 0 be chosen so that Bmo7 C G1 Since f is continuous there is a C gt 0 such that lt C for m E Bmo7i Set 10t E mo and for a given n E N and for j 1 i i i n de ne the sequence via 22mm mo fmj8dsi 13gt Note that 110 6 Bmo 7 for each ji Since f is uniformly bounded one has that ill imj1t moi lfltmjsids lt Citi 14gt thus for iti S 6 I yC one has that mj t E Bmo7i Furthermore since f is continuous one has that each mj t is continuous on 76 ill It will now be shown by induction that CKJ39WH ij1t 93AM S W7 7396 N07 15 ODE Class Notes 12 where K is the Lipschitz constant for f Now by equation 14 one has that equation 15 holds for j 0 Assume that it holds for j 0 n By equation 13 one has that for n 2 1 1mm mt meals 7 rows as Since f is Lipschitz this then implies that ltl lmn1t7 mm s K lmns 7 m1ltsgtl as which by equation 15 with j n 7 1 further yields that CK l l lmn1t 7 g 7 3 ds ml 0 Hence equation 15 holds Now set 00 92t 90 2193171 93139 ill As a consequence of equation 15 the in nite sum is uniformly convergent and since each term is continuous this then implies that mt is continuous on 76 6 Since the sum is telescoping one actually has that t l 39 t 910 P1309110 with the limit being uniform on 766 Since f is uniformly Lipschitz continuous one has that fmj 7gt uniformly on 766 Hence from equation 13 one can conclude that w 0 0tfmsds 76 g t g a 16 The righthand side of equation 16 is differentiable on 766 hence upon differentiating one has that 39m The solution 1 also clearly satis es 1 0 0 In conclusion one has that the solution to equation 16 satis es equation 12 Note that as a consequence of the discussion leading to equation 16 one has the following result Corollary 13 mt solves equation 12 if and only if mt solves the integral equation equation 16 112 Proof by polygonal approximations Eulerls method a numerical method of Oh is given by mn1t mn tn t 7 tnfmn tn1 tn h t E tm tn1l One can de ne a continuous piecewise linear function via mt h mkt t6ltktk1l The goal herein is to show that under the appropriate assumptions on the vector eld Eulerls method converges to a solution of equation 11 ie that there exists a sequence hj with hj 7gt OJr as j 7 00 such that limjnar00 mt hj is a solution to equation 11 De nition 14 Let wk tkeN be a family of functions de ned on I 2 ab The sequence converges uniformly to mt if for every 6 gt 0 there is an Ne such that lmk t 7 lt e for k gt N and t E 1 Proposition 15 If the functions wk t are continuous then mt is continuous 13 Todd Kapitula De nition 16 The sequence wk is equicontinuous on I if for every 6 gt 0 there is a 66 gt 0 such that mkt 7 lt e if t7 s lt 6 for all st E I and k E N De nition 17 The sequence wk is uniformly bounded if lt M for all t E I and k E N Theorem 18 Arzela Ascoli Theorem If wk is an equicontinuous uniformly bounded sequence of functions on I then there is a subsequence wk which converges uniformly to mt on I Remark 19 It is not necessarily true that 1160 7gt mt as k 7 00 furthermore different subsequences may converge to different functions Proof The proof will take place in 6 steps I For 6 gt 0 given subdivide 06 into n equal subintervals 0 to lt t1 lt lt tn 6 so that tk1 7 tk 6n I hn Set MAO MW t tkfmktkv t6 ltkvtk1lv and de ne mt hn mk1t for t E tktk1 Note that mt hn is piecewise linear and continuous In all that follows for 0n 71 e ne wk 2 1tk It must now be shown that by making 6 sufficiently small mt hn E G for all n Let 7 gt 0 be chosen so that Bmo7 C G Since f is continuous there is a C gt 0 such that lt C for m E Bmo7 Thus for t e 0 t1 C6 105 hn flol 75lfquot30l lt Chn 7 Assuming that C6 lt 7 the smallness condition on 6 one then has that 1t1hn 7 mo lt yn so that ml 6 Bmo7 Similarly for t E t1t2 one has new 7 m lt t7t1C 3 Ch lt 1 TL so that 2 satis es mg 7 ml lt yn Thus 27 12 10 S 12 mil 11 10 lt Z ie mg 6 Bmo7 Continuing in this manner it is seen that for t E t tj1 j 0 n 71 1thn7mollt S77 139 1W n so that mt hn E Bmo7 for all t E 0 6 II Since 1t hn 7 mo lt 7 for t 6 06 one has that 1t hn lt 0 7 Hence the sequence 1t hn is uniformly bounded III It must now be shown that 1t hn is equicontinuous so we need to estimate 1t hn 7 18 Assume that s lt t and further assume that they do not occupy the same subinterval There is an ij such that t1ltsgtltt1ltlttjgtlttj1 Now 102 flj t tjfmj7 18 hn 171 8 7tz3971f91i717 and since 1 1111 ti7ti11fmi1 one has that 15 hn 117 s 7 tifmi1 Now mt hn 7 15 hn mthn 7 mj mj 7 mj11 ml1 7 mi 7 15 hn so the identity mk1 7 wk yields 91thn 918 hn t tjf9ij ti 8f91i71 hn fm 1 7 ki ODE Class Notes 14 Since lt C for m E Bmo7 this yields lmthn 7 15 hnl lt t 7 tj ti 7 s j 7 ihnC lt C t 7 3 IV By the Arzela Ascoli Theorem there is a subsequence 1t hnk which is continuous and converges to a continuous mt for t 6 06 Furthermore since 1 0 hn 0 for all n one has that 10 moi Without loss of generality suppose that the full sequence converges to A careful examination of the sequence reveals that for t E 716 tk11 171 mmmn7ma7mvmmm j mx where mj 1tj hn Since the sequence converges uniformly to mt and since f is continuous upon taking the limit it is seen that mt solves the integral equation m mAf Ns no Since is continuous the righthand side is differentiable hence mt is differentiable Taking the derivative yields Mt f93t7 930 910 ie m solves equation 12 VI We need to get a solution on 766 Set 8 7t and consider dy E 7 7fy8 If a solution ys exists then mt y7s is a solution to the original ODE as dm dy 5 7 73 7 rm 7 fltylt7tgtgt 7 mt Thus by the previous steps we have a continuous solution de ned on 76 6 and with it being differentiable on 76 We need to show that fmol The argument in V can be used to show that mt is differentiable from the right at t 0 with the righthand derivative being fmol Similarly the lefthand derivative will be fmol The proof is now complete D Remark 110 As a consequence of the discussion leading to equation 18 one again has Corollary 113 12 Uniqueness Now that it is known that equation 12 has a solution given in equation 16 it is necessary to determine the conditions under which it is unique Afterwards we must then understand how the maximal interval of existence can be determined The following lemma will be crucial in answering these questions Lemma 111 Gronwallls inequality Suppose that a lt b and let 04511 be nonnegative continuous functions de ned on ab Furthermore suppose that either a is constant or a is differential on ab With a gt 0 If E at 3 am 7 wltsgt ltsgtds te aw a then t BWSawd mhytemH Proof See 17 Theorem 11112 or 8 Theorem lllllllll D 15 Todd Kapitula Theorem 112 Taylor7s theorem Let U C R V C Rm be open lff U gt7 V is C1 and m th 6 U for all t E 01 then 1 fm h 7fm Dfmthdtgt h 0 Corollary 113 lff U gt7 V is C1 and m th 6 U for all t E 01 then one has the estimate if91hf93l S CW7 Where C I sup lDfm th Le01 De nition 114 f U gt7 V is Lipschitz if there exists an L gt 0 such that 7 S le 7 yl for all m y E U f is locally Lipschitz if for each mo 6 U and each 6 gt 0 such that Bmoe C U there is an L5 such that if 131 C Bmoe then 7fyl g Lelm 7 As a consequence of Corollary 113 one has the following result Proposition 115 Iff U gt gt V is C1 then f is locally Lipschitz lt is clear that iff is locally Lipschitz7 then f is continuous Hence7 as a consequence of Theorem 11 if f is locally Lipschitz there exists a solution to equation 12 As it will be seen below7 the mild restriction off being locally Lipschitz7 instead of merely being continuous7 actually yields more than simple existence Theorem 116 Uniqueness theorem lff is locally Lipschitz then the solution to equation 12 is unique Proof Since f is locally Lipschitz7 f is continuous hence7 by Peano s Theorem 11 there is a solution Any solution satis es the integral equation mt to 0 fms d5 Suppose there are two solutions7 ml and 2 One has that z am7mm7Ummw4mwma 0 Since f is locally Lipschitz7 there is an L gt 0 and e gt 0 such that as long as 11t 12t E Bmoe then lflt2lt8gtgt 7 S Lim s 7 11sl By Theorem 11 there is a 6 gt 0 such that this condition holds for t E 76 6 One then has that n ma 7 mm 3 L lm s 7 1M d3 0 By Lemlna 111 this yields ma 7 91ti g oeL hence7 11t 12t for all t D 13 Continuity With respect to initial data The following result indicates that under the assumption leading to unique solutions7 the solution set is continuous with respect to variations in the initial data Theorem 117 Continuity with respect to initial conditions Suppose that f is locally Lipschitz Consider the two initial value problems 9391f937910 0 9391f937910 0 Denote the solutions by 10t and mht respectively For each 6 gt 0 there is a 6 gt 0 and L5 gt 0 such that for lhl lt 6 one has imm7mmmw for all t E 766 ODE Class Notes 16 Proof Similar to that for Theorem 116 1 Regarding unique solutions to equation 12 one has the following useful properties Lemma 118 Denote the solution to equation 12 by q5t 10 For any xed to E R 1505 10 4505 i to 1500 loll Proof Set 3 t 7 to so that E 7 3 dt 7 ds hence the form of equation 12 does not change so that both 450210 and q5t 7 to q5t0 10 are solutions Since 450 10 457t0 q5t0 10 by uniqueness they must be on the same trajectory 1 Corollary 119 Suppose that there is a T gt 0 such that the solution satis es 10 With f 10 for all 0 lt t lt T Then q5t T for allt gt 0 ie is a periodic orbit Proof By Lemma 118 the solution satis es w 10 t T T 10 t T 0 E 14 Extensibility Suppose that in equation 12 that f is C1 on the open set C Denote the unique solution by q5t and suppose that J 2 15 is the maximal interval of existence We must now understand what happens to the solution as t 7 1 and t 7gt 7 Theorem 120 Extensibility theorem For each compact set K C G there is a t E J such that K thus 139 t 8G 139 t 8G MUELMK 7 Hug k In particular ifG R then 139 t l39 t 1511 l Hug W l 00 Proof Suppose that there is a compact set K C G such that E K for all t E J Since 0 X K is compact there is an M gt 0 such that S M for all t m E 0 X K Let 8182 6 05 be chosen so that 31 lt 82 One has that i lts2gt 7 sll lf sl as Mls2 7 sll hence 45 is uniformly continuous on 0 As a consequence extends continuously to 0 in particular liming exists and z W 7 me flt ltsgtgtds te 06 0 Now consider the IV 51 f937 935 Bl Since K C G there is a 6 gt 0 such that a solution is de ned on the interval i 575 5 Denote this solution by t and note that by uniqueness for t E 7 6 Set W tam W lw tewwi It is easy to check that 7t solves the original IV on the interval 06 6 Hence the interval J is not maximal D 17 Todd Kapitula Finally reconsider the result of Theorem 120 in the case that G Rni As a consequence of the next result it can be assumed for theoretical purposes that in this situation the solution to equation 12 exists for all time Corollary 121 Without loss ofgenerality iff R gt gt R is continuous then solutions to equation 12 exist for all t E R Proof Let g G C R gt gt R be smooth It Will rst be shown that the trajectories to z39 are the same as those to m Let be a solution to 39m Set z 397 d3 T quot 90158 so that lgq t gt 0 By the chain rule d1 die 1d dt 7 d39rdt igd rl so that dm 9 fm ltgt CT gmf93A 7 Thus is a solution to 1 In particular set 91 2 11 S 1 By Corollary 13 the solution to m is given by t f 0155 t m d 1 0 1 was 3 Which yields the estimate 2 l tllmol1d8lmolltl 0 By Theorem 120 the solution then exists for all t E R 1 15 Examples Herein a few examples are considered Which illustrate the utility of the above theoryi Example Suppose that f R A R is Lipschitz With Lipschitz constant L Since i S lel implies that S lf lel a solution to equation 12 satis es for t gt 0 t 1 mm lmol lfmsl as lmol lf0lt L ms as 0 0 Applying Gronwallls inequality implies that mm s w lf0lte ie the solution is uniformly bounded on 0T for any T gt 0 Since T gt 0 is arbitrary the solution is de ned on 0ooi Reversing time and applying the same argument yields that the solution is de ned on 700 00 In particular When f is linear in m ie ft m Atm then the result of Theorem 120 can be improved via the above argument Corollary 122 Consider it Atm 1t0 mo Where At E Rn is continuous on ab Ifto 6 ab then there is a unique solution de ned on a b ODE Class Notes 18 Proof Let 6 gt 0 be given and suppose that to E a 61 7 e The solution satis es 2 at 7 me Altsgt ltsgt as to Since A is continuous there is an L gt 0 such that lAtl S L for all t E a 61 7 6 Thus after applying Gronwallls inequality one gets WW S lmolele Oly ie the solution is uniformly bounded on a 61 7 e By Theorem 120 the solution exists for all t E a e b 7 6 Since 6 gt 0 is arbitrary the solution exists on a 12 Example Consider 939 ht9I7 1750 3007 Where t0zo gt 0 and g and h are positive C1 functions de ned for t 2 0 and z gt 0 Applying separation of variables the unique solution satis es 1 Ehs dsl Since h is continuous for t 2 0 one has that b hs ds lt 00 0 for any I 2 0 Now assume that 1 dz algae 7ool 19 If limtna 0 for some a 2 0 then 451 0 lim Clix hs ds 140 10 9W to Which is a contradiction Hence under the assumption given in equation 19 one has that limtna gt 0 for any a 2 0 so that the solution exists for t E 0 toll 19 Todd Kapitula 2 LINEAR SYSTEMS In this section we will concerned with solving linear systems of the type i Atm 21 where either At E A At T At or limtnnr00 At A These are the cases which arise most frequently in applications In general a thorough understanding of the solution behavior to equation 21 is necessary before attempting to understand 939 At9 m m 22 see Lemma 24 21 General results Recall from Corollary 122 that if At is is continuous on I I a b and if to E I then for each mo 6 R a unique solution to equation 21 exists for all t E I A set of n linearly independent solutions if it exists is called a fundamental set of solutions Let 3 E R for i n denote the usual basis vectors Pick to E I By Corollary 122 one knows that for each i there exists a solution de ned on I such that 451 to ei If there exist scalars a1 an such that a115105 an nt E 07 then in particular one must have that aiei 0 This is a contradiction hence the solutions are linearly independent If one sets t I 41510 E Rnxn and notes that t t t I 51 MW A4151 14 AQ one has that 39i39 is a matrixvalued solution to equation 21 furthermore it satis es the initial condition to 11 Such a matrixvalued solution to equation 21 is called the principal fundamental matrix solution As a consequence of the above discussion we have the following representation for solutions to equa tion 21 The simple proof is left for the interested student Lemma 21 The solution to equation 21 is given by mt 0310 The next result gives the rst useful property of fundamental matrix solutions eg see 17 Theo rem 66 Lemma 22 Liouvillels Abel7s formula If t is a fundamental matrix solution to equation 21 then detlt1gtt detlt1gtt0effo 69045 As a consequence of Abel s formula one has that if the fundamental matrix solution is nonsingular at one point in time then it is nonsingular as long as it is de ne Corollary 23 t is a fundamental matrix solution if and only if t0 is nonsingular Example Let t be the principle fundamental matrix solution to equation 21 at t to Suppose that 2 lim traceAs d3 2 7M gt 700 tgtoo to It will now be shown that there is at least one solution to equation 21 which is nonzero in the limit t A 00 As a consequence of Liouville s formula lim det t lim e150 in ame 2 e M tgtoo tgtoo ODE Class Notes 20 Suppose that for t I t ei one has that limtnlr00 0 for all ii This necessarily implies that limtnlr00 det t 0 which is a contradiction Hence there exists a j such that limtnlr00 y 0 However the converse is not true Consi er A I diag72 1 Then fol traceAs ds it A 700 however both solutions to equation 21 in this case do not approach zero Now consider t I tB 23 where B E Rnxn is constant and nonsingulari By the product rule d a 39B B At B hence t is a matrixvalued solution to equation 21 which satis es the initial condition 1t0 Bi Such a solution is called a fundamental matrix solutioni Note that as a consequence of the existence and uniqueness theorems all nonsingular matrixvalued solutions to equation 21 are of the form given in equation 23 Finally and as expected the solution to equation 21 is crucial in constructing solutions to equa tion 22 The result of Corollary 13 yields one solution formula however it ignores the effect that the linear solution has on the nonlinear perturbation The below result gives a formulation which is more convenient in applications and which will be used throughout this text Lemma 24 Variation of constants formula Consider 53 A lm ft7917 93 910i lf t represents the principal fundamental matrix solution to equation 21 then the solution is given by z 1t lt1gttmolt1gtt s 1fsmsdsi to Proof It is clear that 1t0 moi Differentiating yields t 9W N010 gttl gt871f87 938 018 gttlt1gtt 1ftv 93W to z Atgtto gt8 1f87938 018 ft793t to Atmtftmti 1 Remark 25 As a consequence of Lemma 2 4 and Gronwallls inequality one may expect that a detailed analysis of the solution behavior for equation 21 will yield de nitive information regarding the solutions to equation 22 22 Equations With constant coefficients In the previous subsection general results concerning solutions to equation 21 were given However knowing that a solution exists is not generally sufficient when attempting to understand the solution behavior to equation 22 In order to answer concrete questions associated with equation 22 it is necessary to understand the solution behavior to equation 21 in great detaili This necessitates that one restrict the form that A is allowed to take In this subsection it will be assumed that A is actually a constant matrixi 21 Todd Kapitula 221 The fundamental matrix solution Consider the partial sum N tn S t I Ani N E N N 7 n o and note that SN0 l for each N 6 N0 For each N one has that SN is smooth in t furthermore since lABl S lAl 3 for AB E Rn one has that for each n E N t T lAL Mi ltl T TL TL which yields the estimate W W w W lt A717 l l n elAT ltl g T N lSNtl S ZlAl 710 24 l In order to continue the following version of the Weierstrass M test is required Lemma 26 Weierstrass Mtest Let SN R gt gt Rn forN 6 No be such that lSNtl S MT for ltl S T The sequence SNt converges absolutely and uniformly for ltl S T As a consequence of the Weierstrass Mtest and the estimate in equation 24 one has that for each fixed T E R the partial sums SNt converge absolutely and uniformly for ltl S T Since T is arbitrary the sums converge for all t E R Furthermore since each SNt is smooth one has that the limit is continuous in t De nition 27 For A E Rn set 00 tn At n 2 lm S t E A i e Nioo N n0 nl At It will now be shown that in the case of constant matrices e is a fundamental matrix solution to equation 21 One has that for each N 6 N0 d SNO ASN1t SN1tA The righthand side follows from the product rule and the fact that AAk AkA for each k 6 N0 By taking the limit of N A 00 and using the fact that the convergence is uniform one has that gem AeA eA A dt Upon noting that eA390 ll one has the following result Lemma 28 The principal fundamental matrix solution to i Am at t 0 is eA It will now be shown that eAt satis es the usual properties associated with the exponential function First let 8 E R be given and consider the initial value problem it Am 10 eAS As a consequence of Lemma 28 and the discussion in Section 21 one knows that the unique solution is given by eA eAS Now set t I eA5 and note by the chain rule and Lemma 28 that Li A IJ 110 e145 ODE Class Notes 22 By uniqueness one can then conclude that eAst eAteAs Upon switching s and t one then gets that eAst eAts eAseAt hence eAst eAteAs eAseAt 25 Note that upon setting 8 it in equation 25 one gets that 71 6A1 eiAt Now suppose that B E Rnxn is such that AB BAl One then has that BSNt SNtB hence upon taking the limit one has BeA eA Bl By the product rule eAteBt AeAteBt eAtBeBt A BeAteBt Note that the above argument also yields d BL AL AL BL e e ABe e However by Lemma 28 one also has that 6ABt A BeABt hence the by uniqueness of solutions one has that eABt eAteBt eBteAt Finally suppose that B is nonsingularl Since BilAnB B IABYL for each n 6 No one has that N tn 71 7 i 71 n B SNtB 7 20mm AB l 7 Taking the limit of N 4 00 and using De nition 27 gives BileAtB eB iABt The above argument can be summarized as following Lemma 29 Suppose that AB E Rn and that st E R Then a eAt71 67A b eAst eAseAt eAteAs c if AB BA then eAB eA eB eB eA d ifB is nonsingular then BeA B 1 eBABBI We will now consider some special cases in which eA can be easily computed Lemma 210 Suppose that A PAP I Where A diag1uln and P v1 1 with Avk Akvk Then eA Pdiage 1 l l eAquot P 1 23 Todd Kapitula Proof By Lemma 29d one has that 6A PeAtP71 hence it is enough to compute eA Since Ak diaglf A176 one gets that 00 oo M7316 A tk eAz dug T Z dlage117 yeknz D 160 160 Lemma 211 IfA A1 N then eA eA eN Proof Since 1N N11 one can immediately apply Lemma 29c Implicit in the calculation is the identity eMn emll Finally suppose that 01 Aal7bJ Jlt710gt where ab E R The eigenvalues of A are a i ii furthermore by Lemma 211 one has that eA eate bh It can be checked that 0quot n bt 2n 0quot n bt 2n1 204 27 7724 2n1l 7th 0g 2 1 00 2 bt 7 bt 7 Zlt71gtnL Zlt7D7Llt n70 2n 1 770 2n 7 cosbt 7 sinbt 39 7 sinbt cosbt hence At 7 at cosbt 7 sinbt e 7 e sinbt cosbt As a generalization suppose that A E R2ngtlt2n has complex eigenvalues Aj aj ib39 for j 1 n If the eigenvectors wj uj ivj are such that 111 un v1 vn is a linearly independent set then as a consequence of 15 Chapter 16 it is then known that for PI vlul vnun one has P lAP A 397 al 7121 an 712 A7d1agltlt b1 a1gtlt bn an 26 As a consequence of the above discussion A 7 a t cosb1t 7 sinb1t an cosbnt 7 sinbnt e 7 dlag lt6 1 lt sinb1t cosb1t 7 e sinbnt cosbnt 27 Upon using Lemma 29d the following lemma has now been proved Lemma 212 Suppose that A PAP I Where A is given in equation 26 Then where 6A PeAtP717 where e is given in equation 27 ODE Class Notes 24 Example Suppose that 74 0 0 A z 0 3 72 0 1 1 The eigenvalues and associated eigenvectors are A1 7411 100T A2 21 w2 01 i1T Setting P v1 1111102 Re 102 yields that 74 0 0 A P lAP 0 2 71 0 1 2 Thus e4 0 0 em 0 e2 cost 7e2 sint 0 e2 sin t e2 cost and eA PeA P I 222 The Jordan canonical form The results of Lemma 210 and Lemma 212 required that one be able to diagonalize A in a particular manner If A is symmetric or if more generally the eigenvectors form a basis for C then A can be diagonalized in such a way However there are special cases Which are typically bifurcation points in parameter space for Which the diagonalization assumption leading to Lemma 210 and Lemma 212 breaks down It is this case Which Will be covered in this subsection Herein it Will be assumed that all of the eigenvalues of A are realvalued The case of complex eigenvalues is covered in 15 Section 18 De nition 213 The spectrum of A E Rn is given by 0A A E C detA 7 All 0 The multiplicity of A E 0A is the order of the zero of detA 7 All 0 De nition 214 Let A E 0A have multiplicity p Let v1vm for 1 S m S p form a basis for kerA 7 All Then a m is the geometric multiplicity A ie mg A m b p is the algebraic multiplicity of A ie maA m A E 0A is simple if mg A maA 1 and A E 0A is semisimple if mgA maA 2 2 Remark 215 If mgA maA for each A E 0A ie if each eigenvalue is semisimple then A is diagonalizable De nition 216 Let A E 0A be such that arn p For k 2p any nonzero solution 1 of A 7 All 0 With A 7 Allk 1v 0 is a generalized eigenvector Remark 217 Note that kerA 7 Ally C kerA 7 Allj1 for anyj E N In order to better illustrate the above ideas consider the following generic example First suppose that A 1 0 1 Alt0 AgtA1lt0 0 One has that A E 0A satis es mgA 1 and maA 2 The eigenvector is v 10T and the generalized eigenvector is w 01 T To generalize for a givenj 2 2 let N 6 1Rij satisfy 1 Z1Z13971 Ngmz m 7 28 0 otherwise 25 Todd Kapitula It can be checked that N is a nilpotent matrix of order j ie Nj 0 and Nj 1 0 hence 141 tn eN 7N nl 710 Note that in this case the entries of eN are polynomials of degree no larger than j 7 1 For the matrix A All N one has that A E 0A with mg A 1 and man j Finally from Lemma 211 one has that 141 n eAt eAzeNz em nl 710 In order to put a matrix in Jordan canonical form the general idea is to rst build chains of eigenvectors using the above ideas This construction will not be carried out herein eg see 15 Section 18 and the references therein as the procedure is quite technical and all that is necessary in the subsequent sections is the nal result As is seen below the key to 39 quot quot quot a quot quot matrix is to use nilpotent matrices of the form given in equation 28 As a result of the above discussion note that for each Jordan block in the statement of Theorem 218 one has mg A 1 Theorem 218 Jordan canonical form Suppose that A E Rn has real eigenvalues A1 An There exists a basis ofgeneralized eigenvectors v1 1 such that With P 2 v1 vn one has that PilAP diagBl 3 Where the Jordan blocks are of the form Bj All N 6 R2 for some 1 S E S n and N is given in equation 28 As a consequence of Theorem 218 Lemma 210 Lemma 211 and Lemma 212 one has the following result Theorem 219 Every entry of eAt is composed of linear combinations of ptem cos t and ptem sin t Where A a i E 0A and pt is a polynomial of degree no larger than n 7 1 Example Consider 71 1 72 A 0 71 4 0 0 1 Here mg71 1 and ma71 2 while mg 1 ma1 1 It can be checked that with v1 021T 02 100T v3 010T upon setting P v1 v2 v3 one has P lAP diagBlBg B1 1 32 lt 3 j One then has that 223 Estimates on solutions Now that the construction of the principal fundamental matrix solution is understood it is time to see how one can estimate the solution behavior The following preliminary result is rst neede Proposition 220 For each 6 gt 0 and each j E N there exists an Mj e gt 0 such that l 66 tngltjegte Mltjegtlt gtj for all t 2 0 ODE Class Notes 26 Proof Set gt I tje Since gt is continuous and satis es gt A 0 as t A 00 there exists a Mje gt 0 such that gt S Mj7 e for all t 2 0 The upper bound is found by nding the maximum of W D Now7 by Theorem 219 each entry of eA is composed of linear combinations of terms like ptem cos t and ptem sin t7 where a i E 0A and pt is a polynomial of degree no greater than n 7 1 The next result then immediately follows from Proposition 220 Lemma 221 Set O39M I maxReA A E aA For each 6 gt 0 there exists an Me gt 0 such that eAt S M6eUM t for all t 2 0 If all A E 0A are semisimple7 then in Theorem 219 one can set pt E 1 ln Proposition 2207 if one sets j 07 then one has 6 0 hese observations yield the following re nement of Lemma 221 Corollary 222 If all A E 0A are semi simple then one can set 6 0 in Lemma 221 Lemma 221 sets an upper bound on the growth rate of leA l The below results sets a lower bound Note that it does not depend on the multiplicity of the eigenvalue with minimal real part Lemma 223 Ham 2 minReA A C aA then eAt 2 eamt for all t 2 0 Proof Let A E 0A be such that ReA am and let 1 be the associated eigenvector One has that eA v eh7 so that At 1 le l Gum W The result now follows from the de nition of the matrix norm 1 Let us now re ne the above estimates ln particular7 we wish to nd invariant subspaces which have proscribed behavior for solutions residing in them Let A E Rnxn and let A E 0A be such that mg A m and ma A p 2 m As a consequence of the reduction of A to Jordan canonical form one has there exists a basis of generalized eigenvectors UN such that with vo I 07 m A7A1vji vj1i i1mj1ai 20417 i1 De nition 224 For each A E 0A set EA 2 spanvji The subspace EA is the called the generalized eigenspace of A corresponding to the eigenvalue Proposition 225 EA is invariant under multiplication by A ie AEA C EA Proof Let 1 6 EA be given7 so that v j 0N1 for some constants CM By linearity one has that Av ZenAv ZenAij Dru A1 Zamoru ij quot 1 1 Thus7 A1 6 EA D As a consequence of the de nition of eA and Proposition 225 one has the following result Corollary 226 eA E C EA Proof By Proposition 225 AEA C EA7 which by an induction argument yields that AkE C EA for each ls 2 1 The result now follows from the series representation for eA and using the fact that EA is a closed subspace D 27 Todd Kapitula The following spectral subsets given below will be used to decompose R De nition 227 The stable spectrum and subspace are given by 05A I A E 0A ReA lt 07 ES 2 GBEAJ Aj E 05A7 the unstable spectrum and subspace are given by aquotA I A E 0A ReA gt 07 Equot I GBEAJ Aj E a A7 and the center spectrum and subspace are given by 06A I A E 0A ReA 07 EC 2 GBEAJ Aj E 06A Proposition 228 One has that a R ESEBE GBEC b dimEsgt gtC is the number of generalized eigenvectors in the basis for Esgt gtc respectively 0 eA EN gtC C Esgt gtc respectively Proof By construction Es7 E 7 and EC are mutually disjoint The statement of a then follows from the fact that the generalized eigenvectors form a basis of R Since AE C EA for each Aj E 05gt gtCA7 parts b and c follow from Corollary 226 and linearity D Armed with Proposition 2287 one can use Lemma 221 and Lemma 223 to describe the behavior asso ciated with solutions residing in each invariant subspace Esgt gtC Theorem 229 If mo 6 Es then there exist positive constants c lt a and m M 2 1 such that me m mO S eA mO S Me c mO While if mo 6 E then there exist positive constants c lt a and m M 2 1 such that me md S eA mt S Meaqmo Finally if mo 6 EC then there exists a k 6 N0 With 0 S k S n 7 1 such that mlmol leA mol M1ltlklmol Remark 230 If all A E a A are semisimple7 then k 0 hence7 in this case solutions residing in EC are uniformly bounde Proof The result will be proven only for mo 6 E57 as the other proofs are similar Set a I minReA A E 05A7 0 2 maxReA A E 05A7 and note that a S 0 lt 0 By Proposition 228 eA Es C Es furthermore7 by Lemma 221 and Lemma 223 for each 0 lt e lt 04 there is an Me such that me f tlmol g eA m g Mee0 t mol 1 One of the implications of Theorem 229 is that the behavior for solutions residing in E5 is exponential in nature Solutions in the unstable subspace exhibit growth for t 2 07 while those solutions in the stable subspace decay for t 2 0 The behavior of solutions in the center subspace is unknown without more detailed information7 and all that can be said is that any temporal growth is polynomial in nature One can summarize in the following manner By Proposition 228 one can write the initial data as s c u scu scu momomomo to CE Using linearity then yields eA mo eA mE eA mg eA mB 29 The solution behavior associated with eA mE C quot is given in Theorem 229 The result of equation 297 along with the de nitions given in De nition 227 associated with the various subspaces7 can be summarized in the following de nition ODE Class Notes 28 De nition 231 Consider it Am The critical point x 0 is a o sink attractor 0A 05A 0 source repeller negative attractor 0A a A o saddle unstable saddle aA 05A U a A with 05gt A f g If aCA 2 then the system is hyperbolic and the associated ow 45410 eA mo is called a hyperbolic ow 224 Linear perturbations stable Consider equation 22 under the assumption that f is smooth enough to ensure unique solutions Recall the result of Lemma 24 Assuming that At E A by using the results of Lemma 28 and Lemma 29 one can reformulate the solution as t mteAquot mo eAquot5fsmsds 210 to Before continuing we need the following de nition which characterizes the behavior of solutions near critical points ie zeros of the vector eld for equation 22 De nition 232 Consider equation 22 A critical point a is stable if for each 6 gt 0 there is 6 gt 0 such that if mo 6 Ba6 then mt E Bae for all t 2 0 The critical point is asymptotically stable if it is stable and if limtnsr00 mt a If the critical point is not stable it is unstable Remark 233 Consider the linear system 391 Am Upon applying the results of Theorem 229 it is not dif cult to show that m 0 is o unstable if a A g o stable if aA 05A U a A and all A E a A are semisimple o asymptotically stable if aA 05A Consider equation 22 under the assumptions that At E A and m m Ban iBti at lt oo 0 First suppose that 0A 05A U a A and that each A E a A is semisimple It will be shown that the solution which is given in equation 210 is bounded By the assumption on 0A one can apply Theorem 229 and conclude that there exists an M gt 0 such that ieA i S M hence the solution satis es the estimate 1 WW S Mimol MlB8l l938ld5 to Upon using Gronwall s inequality one gets that t S M mo eMLO lB5ldS7 so by assumption there is a C gt 1 such that S Cimoi for any t 2 to In particular this estimate shows that m 0 is stable Now suppose that 0A as A so by Theorem 229 that there exists an Ma gt 0 such that ieA i S Me As above one then has that t imlttgti Me quot imoi Me quot5iBsi imltsgti as to 29 Todd Kapitula Set yt I lmtlem so that the above can be rewritten as lytl SMlytolL MlBSHy8ldamp As above there is a C gt 1 such that S Clytol which implies that mm 3 Clmole mt wh Hence 1 0 is asymptotically stable 225 Linear perturbations unstable Finally suppose that 0A 05A U a Ai In this case it will be shown that under the assumption tha 00 lBtldtltoo as t 7gt ioo the system will have the same behavior as the unperturbed systemi The following discussion has as its inspiration the work of 5 Lecture 4 Before continuing the following preparatory theorem is needed Theorem 234 Banach7s xed point theorem Let X be a complete normed vector space and let D C X be closed Let T D gt7 D be such that for all u v E D 7 TUH S LHu 7 11H 0 lt L lt1 in other words assume that T is a contraction mapping There is then a unique uquot E D such that Tu u Proof Let uo E D be given and for n E N de ne the sequence via un1 First note that via an induction argument one gets Hun1 unll llTun Tun71ll S Lllun un71ll S Lnllul uollA This then implies that for each k E N n L 17L 171 171 171 llunk unll E Z llunj71 unjll S ZLnHllul uoll Lnllul uoll EL S llul uoll j 10 10 Thus is a Cauchy sequence and since X is complete and D is closed one has a u E D such that un 7gt u Since T is continuous one then has that Tu u Lastly uquot is unique for if there exists another xed point 1quot then Hu v ll HTW Tvll S LUV H which is a contradiction as L lt 1 1 Remark 235 An application of Theorem 234 allows for a much easier proof of Theorem 116 for the system it fm 10 moi 211 Let eL E R be such that for all uv E Bmo26 7fvl S Llu 7 UL Now set l c supmm a e Bltmo2egt 6 2 mg i and de ne X 1 00li575lR 7 WM 1 sup WW7 M55 ODE Class Notes 30 and D 2 m E X sup 7 10 g 26 M36 If one now considers the mapping T D gt gt X given by Tm mOOtfmtdt ltl g 6 then it can be shown that T satis es the conditions given in Theorem 2 34 hence there exists a unique solution to equation 211 The remaining details will be left for the interested studentl Let P E Rn be such that P lAP A diagAsA where As quot 6 R Sgt Xnsgtn with ms nu n and 0As quot 05gt As quotl Setting 1 Py transforms the original system to N N y AyBt317 Bt P 1BtPl 2 12 De ne the projection operators Us by HE 7 diagus 0 nu 7 diagw in where 115 E RWSWL VWSWL X Note that the projection operators satisfy the properties HsHu HuHs 0 H3 H5 Us H 11 2 13 furthermore Ewe eA Hw 214 As a consequence of Theorem 229 one has that there exists a Ca E R such that eA m g Ce m t2 0 leA Hul g Ce tg 0 215 Choose to E R suf ciently large so that 00 N 90 lBtHdtlt1 to and set X 1 00lt0700R 7 NW 1 SUP WW7 220 De ne the mapping 2 N 00 N Ty eAquot5HsBsys ds 7 eAquot5HuBsys as 216 to t It is not dif cult to show that T X gt gt X with S OllyH and that for any y1y2 E X 00 N m 7 w s 0 lBsHy1s7 y2sl as O S 9 7 ml Consequently T is a contraction map on Xi Now let yo 6 R be given so that Hsyo yo and consider the integral equation 1N eAquot yo HM 2717 As a consequence of Theorem 234 there is a unique solution in X to equation 21 furthermore it is easy to verify that any solution to equation 21 is also a solution to equation 21 with the initial condition 00 N we 7 yo 7 eAquot5gtHuBsgtysgtds to 31 Todd Kapitula Note that there is then a onetoone mapping between bounded solutions of the unperturbed problem and those for the perturbed problemi Now it must be shown that the solution yt A 0 exponentially fast as t A 00 From equation 215 one has that for t 2 to7 lTytlem S Ctlf s ly8lemds00 eltamltt gtl13lt8gtl ly8lemd8 to L S 9llytemll Consequently7 for t 2 to the solution to equation 21 satis es MANGO S Cemolyol lTytlem S Cemlyol Ollyte ll7 Lei7 llytemll S Ceatolyol9ll11lttgtemll This necessarily implies that 1 9Hyt6mll S Cem jlyolv which in turn yields that mm lyole quot l ln conclusion7 it has been demonstrated that there is a oneone mapping between exponentially decaying solutions for the perturbed problem and those for the unperturbed problemi Finally consider equatlon 212 under the time reversal 7 I it One then has d 7A 7 B 7 I 7i y y y dT Since 07A 70A the above argument shows that there is a oneone mapping between exponentially decaying solutions as 7 A 007 iiei7 as t A 700 for the perturbed problem and those for the unperturbed problemi Note here that yo 6 Eur 226 Nonlinear perturbations In the previous example the behavior of solutions to equation 22 in the case of linear and asymptotically zero perturbations was considered The next result deals with nonlinear perturbations which are small in a neighborhood of the origin Theorem 236 Consider equation 22 under the assumptions that At E A and lf t7 0lml2 for lml S 6 Suppose that 0A as A There then exist constants C7M7a such that if lmol S M then the solution satis es S Ce u quot lmol for allt 2 to In particular 1 0 is asymptotically stable Proof Since 0A 05A7 by Theorem 229 there exists a C 2 1 and A gt 0 such that leA l S Ce Mi Since lftml 0lml2 there exists a k gt 0 such that lftml S klmlz Fix 6 gt 0 so that e lt 6 and Seta7Cke andpeCandnotethatagt0and0ltpltelt6i If lmol lt M there exists a 739 gt to such that the solution mt satis es S e on the interval I I t E R z to S t S 739 This implies that for t E I one has that WHOM S kl91lttgtl2 S 764w By equation 210 the solution is given by t m 040220 were mltsgtgt as to ODE Class Notes 32 which yields that as long as t E I z S Ce quot lmol Cket e quot5lmsl ds 0 Rearranging the above inequality gives 2 e quot lmtl S Clmol Cke e 5 t lmsl ds to which by Gronwall s inequality yields WWW Clmol60k5quot or nally S Clmole quot lf lmol lt a then the above estimate yields that S e for all t E I By the extensibility Theorem 120 one then has that 739 00 furthermore m 0 is stable Since a gt 0 z 0 is asymptotically stable 1 Now consider 9391 at 910 9o 218 where f E CWR for some k 2 2 Suppose that fa 0 Upon setting y m 7 a equation 218 becomes gy where gy I fa Note that 90 0 hence without loss of generality one can always assume in equation 218 that a 0 Now by Taylorls theorem one can write f93 A93 91 where A I Df0 and 1 x fm 7 Am Dfsm 7 A d3 1 0 Since is smooth by the Mean Value Theorem lDf891 Al S lsml sup lD2fTml S M sup lD2fTml 01 760gt1l furthermore since D2f is continuous there exists a 6 gt 0 and a constant k gt 0 such that sup lD2fTml S ls m E B06 01 Thus 0lml2 for m E B06 Applying Theorem 236 yields the following result Corollary 237 Consider equation 218 Where f R gt7 R is smooth With f0 0 Set A I Df0 lfaA 05A then there exists an a gt 0 and a neighborhood U of m 0 such that if mo 6 U then S Clmole m for all t 2 0 Now consider equation 218 under the time reversal s I 7t One than has d 7A 7 I 7 m 1 x d3 Since 07A 70A Corollary 237 now applies for s 2 0 ie t S 0 Corollary 238 Suppose that 0A a The result of Corollary 237is true fort S 0 33 Todd Kapitula 23 Equations With periodic coefficients Herein we will consider equation 21 under the condition that At is continuous and T periodic7 ie7 At T At for some T E R3 In order to understand the issues involved7 first consider 139 atz7 at T at asds The fundamental matrix solution is given by t efti Setting a 2 OT as ds7 pt I Otas 7 1 d3 yields t Pte0 7 where Pt 2 e170 Now7 z tT pt T 0 as 7 1 d3 t as 7 1 d3 T W ltaltsgt 7 a as W so that the fundamental matrix solution is the product of a periodic function with an exponential function The initial goal is to show that this property is true for systems We need a preparatory lemma Lemma 239 If C E Rn is nonsingular7 then there exists a B E Cnxn such that e3 C Proof Let J be the Jordan canonical form of C7 ie7 P 1CP J If eK J7 then ePKPi1 C hence7 it can be assumed that C is in canonical form Let A17 7 Ak E 0C have multiplicities n17 7 nk One has that C diagC17 7 Ck7 where each Cj E Cnixni with Cj Ajll N7 where N is nilpotent of order nj Since C is nonsingular7 M f 0 for all j Motivated by the fact that 00 In ln1 1 271 1Z7 lt 17 711 n71 71W N n 371nAjisj7 37 Zglt7 n X n1 J The sum is finite because N is nilpotent As a consequence of Lemma 29c one has e37 AjeSV It can be shown 87 p 6162 that l S i 7N39 e Aj 7 hence7 e37 Cj Upon setting B I diagB17 7Bk7 one has that e3 C D Remark 240 The matrix B given in Lemma 239 is not unique7 as one has the identity eB2Z7rin eBe2Z7rin CL Z 6 Z As the following result shows7 the decomposition of the fundamental matrix solution given in the beginning of this section for a scalar problem also holds for systems Theorem 241 Floquetls theorem Consider equation 21 Where At E Rn is continuous With At T At If t is a fundamental matrix solution then there exists a B E Cnxn such that ltIgtt PteB 7 Pt T Pu ODE Class Notes 34 Proof Since At is Tperiodic if t is a fundamental matrix solution then so is t T By the uniqueness of solutions one then has that MDqmay By Lemma 239 there is a B E Cnxn such that T eBT which yields tJr T teBTi Upon setting Pt I te B one has that POD JWW EP i n De nition 242 The monodromy operator for the fundamental matrix solution given in Theorem 241 is given by eBTi A E 0eBT is known as a Floquet multiplier and M 6 03 is a characteristic exponenti If M is a characteristic exponent then A e T is a Floquet multiplieri Since B is not unique the characteristic exponents are not unique However the Floquet multipliers are unique as these are given by 0 Ti As is the case for equation 21 when A is constant one has that 03 plays a signi cant role in the behavior of solutions Lemma 243 lfM 6 03 then there exists a possibly complex solution to equation 21 of the form e pt Where pt is T periodic Proof Let t be the principal fundamental matrix solution at t 0 so by Theorem 241 one has that t PteB with P0 11 Since M 6 03 there exists a 1 such that 31 Mvi Thus mt I tv e Ptv which proves the result upon setting pt I Ptvi 1 Note that the solution given in Lemma 2 43 satis es the identity mt T Amt where A I e T E 0eBT is a Floquet multiplieri An induction argument then yields that mt 7LT A mt for any n 6 Ni Now consider the sequence wk where wk 2 1kT Akm0i 1f mj 0 for somej 6 N0 then one has a jTperiodic solution to equation 21 The argument leading to this assertion is similar to that leading to Corollary 1 19 and is left to the interested studenti Since the solution is uniformly bounded on 0 T if one wishes to understand the dynamics associated with equation 21 it is sufficient to look at the behavior of this sequence However this is equivalent to looking at the sequence 1f lAl gt 1 then lAkl lAl A 00 exponentially fast hence the solution x 0 to equation 21 is unstable 1f lAl lt 1 then 1 gt lAlk A 0 exponentially fast so that the solution x 0 to equation 21 is stable Now suppose that lAl 1 which implies that A ei27r for some 9 6 01 1f 9 is rational ie 9 104 for 104 6 N relatively prime then there is the periodic sequence given by 1 ei27r9 ei47r9 ei2q717r9 so that m moi Thus there exists a qTperiodic solution to equation 21 1f 9 is irrational then the orbit is dense on the circle lAl 1 and the orbit is uniformly bounded but not periodici Lemma 244 Let A1 i i i An 6 0eBT Then a lAjl lt 1 for allj implies that m 0 is asymptotically stable b lAjl gt 1 for somej implies that m 0 is unstable 0 lAjl S 1 for allj With lAjl 1 being semi simple implies that m 0 is stable Proof Set 1 Pty which gives 39 T53 P y P197 iiei 39 QP 4M PWPWWA Since Pt te B one has that Pt AtPt 7 PtB so that upon substitution 3yi As a consequence of Theorem 229 the behavior of yt is determined by 03i Now ifM E 05gt gtC3 then A e T E 0eBT satis es lAl lt 1 lAl gt 1 lAl 1 respectively Since Pt is continuous and Tperiodic the result follows E 35 Todd Kapitula As a consequence of Lemma 244 one knows that the dynamical behavior is determined by a 39T However one cannot usually directly compute the Floquet multipliers as the explicit form of the fundamental matrix is generally unknown As the following result shows one can still get some information regarding the multipliers Lemma 245 If Aj el iT are the Floquet multipliers then a H Aj ejoTtraceAsds j1 b uj OT traceAs d3 mod Proof Upon using Abel7s formula in Lemma 22 and assuming that P0 ll PT det T det eBT elOT aceA5 d5 Part a follows from the fact that n det eBT H A j1 while part b follows from A JELI M 1 ES 1 x H 231 Example periodic forcing For the rst example which illustrates the utility of Floquet theory consider i Atm bt 219 where At E Rn and bt are Tperiodic One has the following result concerning the existence of Tperiodic solutions to equation 219 Lemma 246 Let t be the principal fundamental matrix solution for equation 219 Ifl a T then there exists a unique T periodic solution Proof By the variation of constants formula the solution to equation 219 is given by z w mm t s 1bs d3 0 In order to have a periodic solution one must have that 1T 10 Upon some algebraic manipulation this yields T 11 7 T zo 0 T s 1bs d5 220 If A E a T then 1 7 A 6 011 7 T Hence ifl a T then 11 7 T is nonsingular and 0 is then uniquely given by T we 11 7 T 10 T s 1bs d5 D In general of course it is dif cult to compute the Floquet multipliers However if At E A then one has that if A E 0A then eAT E 0eAT The following result then follows immediately from Lemma 246 ODE Class Notes 36 Corollary 247 Consider equation 219 under the condition that At E A If there exists no A E 0A such that 2 Z A 1 z e Z 221 then there exists a unique T periodic solution Remark 248 The condition in equation 221 is automatically satis ed if 06A gr Even if the condition ofl a 39T in Lemma 246 is removed it may still be possible to nd periodic solutions to equation 2 19 Upon using the fact that t is nonsingular equation 220 can be rewritten as T ltIgt 1T 7 1090 s 1bs as 222 0 lf 1 E a T then a solution to equation 2 exists if and only if T L s 1bsds e ker gtT 1 7 MT 223 0 Now suppose that At E A and further suppose that T 27L Assume that i E 0A is a simple eigenvalue and that ii 0A for any other Z 6 Nli Using Theorem 2 18 write P lAP A where there exist block matrices Bl i i Br such that A diag7JBlHiBT J Zlt 71 1 gt i The change of variables y P711 allows one to rewrite equation 219 as y Ag Ct Ct I P 1btl Since T T e QWA 7 llT diag 067271 7 liue 27rBT 7 ll note that eBT eBT one has the righthand side of equation 2 23 satis es ker e QwA 7 MTi spane3i i i en Thus there exists a 27rperiodic solution to equation 219 if and only if 27r ej e Ascsds0 j12i 2 24 0 Alternatively since A PAP l one has that ATP T P TAT where B T I Bil In the original variables equation 224 can then be rewritten as 27r P Tej e A5bs d3 0 j12i 2 25 0 A geometric interpretation of equation 225 is that the righthand side of equation 22 is orthogonal to spanP el P7 32 ie the eigenspace of A associated with the eigenvalues iii 37 Todd Kapitula Example Consider i DJQI ft ft 27f By Corollary 247 there is a unique 27rperiodic solution if Wei GNU Now suppose that the forcing is resonant iiei w E No The righthand side of equation 22 is given by l 27f sinws d t w 0 S w cosws 3 hence upon applying equation 225 there Will exist a 27rperiodic solution if and only if 27r 27r sinws d3 cosws d3 0 226 0 0 Otherwise the resonant forcing Will produce unbounded growthi Note that if one Writes in a Fourier series ie 00 oo ft f0 2 an cosnt Z In sinnt n1 n1 then equation 226 is equivalent to requiring that al b1 0 232 Example the forced linear Schrodinger equation Consider 1 1t i4 7 wt 610W 100 T W With the boundary condition q7N7rt qN7rt for some N E N and for all t 2 0 Upon using a Fourier decomposition and setting 00 417 t Z qntemN7 n7oo one sees that for each n E Z 2 iLjn 7 Gang eptqn an I w Upon Writing 4 2 un ivn one then gets the ODE 39mn Antmn Ant I JHt Where I z 7 1 1 Hteanept1i One can understand the dynamical behavior of mnt through the use of Floquet theoryi Set 2 n t 2 ant 6 p8 ds 0 Since A7t dt Ant Ant A7t dt as a consequence of Lemma 2i9b a fundamental matrix by solution is given 7 Us n 5 5 7 cos nt sin n t Tn 7 e A d 7 lt isin nt cos nt gt The Floquet multipliers A are the eigenvalues of n T and are easily seen to be given by Ai cos n T i i sin n T ei9fT n ODE Class Notes 38 where 1 T 6W an M 15 T pltsgt as 0 The two linearly independent solutions are given by mic aws pat m T hence one has that mnt will be ZTperiodic if 27f j an6 T2 where j E Z are relatively prime otherwise the motion will be bounded but quasiperiodicl Note that since an contains the free parameter w one can always guarantee that at least one of the Fourier coefficients will be periodicl The full solution to the linear problem will be bounded but quasiperiodicl Remark 249 It is an interesting exercise to attempt to solve the linear problem 1 141 g 7 mg spa gem t 4N7n A What restrictions could you make to make the problem more tractable 233 Example linear Hamiltonian systems In many applications systems of the type i JHtm HtT Ht 2 27 arise where Ht E R2ngtlt2n is symmetric and J is nonsingular and skewsymmetric iiei JT iJl ln applications one often also has that J71 J so that JJ fill One such example was given in Section 23 i Let t represent the principal fundamental matrix solution to equation 22 Since JHtT 7HtJ the adjoint problem associated with equation 22 is given by y Htin 2 28 Assuming that J2 ill one gets that d 2 Jm iJ HtJJm HtJJm hence if m solves equation 22 then Jm solves the adjoint problem equation 228 Thus J is a solution to the adjoint problem so that the principal fundamental matrix solution to equation 228 is given by t 7J tJi Now it can be checked that another solution to equation 228 is tVTl Uniqueness then implies that ltIgtt 1 7J39igttTJ IdgttTI 1 iiei 39i39T is similar to ll Thus for M E a T one has that ifl E a Tl Since T E RQMQ one also has that if M E a T then Mquot E a Tl This argument yields the following lemma Lemma 250 Consider equation 227 and suppose that J E R2ngtlt2n is skew symmetric With J71 JT lf t is the principle fundamental matrix solution then M E a T implies that lM M UV 6 a T 39 Todd Kapitula Now suppose that Ht A ePt PtT Pt where both A and Pt are symmetric By following the argument leading to Lemma 250 it can be seen that if A E 0JA then 7Ai E 0JAi If one assumes that 0A 05A or 0A a A then it can be shown that one actually has 0JA C ilR 11 12 Under this assumption let iiuj E 0JA for j l i i ni When 6 0 the characteristic multipliers are given by i i39 T 39 pj e W7 1Hini These multipliers satisfy l and are distinct if 2 Mja O mod 1Hin Mjimcfo mod jib 229 Since the multipliers vary continuously under perturbation if equation 2 29 holds when 6 0 one has by Lemma 250 that there is an 60 gt 0 such that for the perturbed problem the multipliers are simple and satisfy lpl 1 for 0 S e lt 60 Hence as a consequence of Lemma 244 one has that the trivial solution is stable for the perturbed problemi Example Consider the following variation of the problem given in Section 232 1 uh i4 7 wt 7 2 Pt003147 100 T 7 00 with the boundary condition 47N7r t qN7r t for some N E N and for all t 2 0 Upon setting 00 417 t Z qnte N7 and using the fact that 1 cosz ew e7 one sees that 7 l m inxN COSIq 7 2 Z LiniN qnNe 7700 Hence for each n E Z 1 7 mm 5pt4niN 4nN7 an I w Upon writing 4 2 un iv and setting I lt ff 1 gt one gets for mu 2 un vnT the ODE 27 7Jan19n eptmnN mnN 2 30 For xed n set I yj 1 mnjN7 51 anjN an 2 1 The system equation 230 can then be rewritten as yj JWJ39MJJ39 51003113971 yj1 231 Since j E Z at this point equation 231 is an in nitedimensional ODE Now truncate by supposing that for some M 2 1 one has that yik E 0 k 2 M 1 Under this restriction equation 231 then becomes 4M 7 1 dimensional and can be written as 139 MD eptBy 2 32 ODE Class Notes 40 where y ySMlHyMT ll 2 diagIlHJ D I 7diag MlHl Ml and B is symmetric and satis es Emil ill and is zero elsewhere with an obvious abuse of notation When 6 0 one has that ii j E 0llD j 7MHlMl Using equation 2 29 and applying the theory preceding this example it is then known that all solutions will be bounded for 0 S 6 ltlt 1 if o mod j7MmM Bji w o modg jy kl It is an exercise to give precise conditions on w n N such that the above holds true 234 Example Hill39s equation Herein we will consider a simple example problem which is surprisingly dif cult to analyze eg see 14 Consider i atz 0 233 where a R gt gt R is a continuous Tperiodic function A simple rescaling argument yields that without loss of generality one may assume that T 7L Herein the focus will solely be on developing a stability and instability criterionl It will rst be shown that if 0 S 7r as ds S at 2 0 234 then the trivial solution is stable In other words it will be shown that equation 2 34 yields that the Floquet multipliers associated with equation 233 have modulus equal to unity see Lemma 244 Note that after writing equation 233 as the rstorder system 391 Atm with At1lt 73 1 one has that trace At 0 hence by Lemma 245 the Floquet multipliers satisfy AlAg 1l 235 If A1 A2 6 R with A1 A2 then by equation 235 one has that without loss of generality Mll gt 1 so that the trivial solution is unstable If A1 A2 1 then there exists a solution zp such that IP t 7r zp t whereas if A1 A2 71 then there exists a solution zp such that IP t 27f zptl In either case by using reduction of order a second linearly independent solution is given by 12t utzp t where W 255 0 This solution may or may not be unbounded as t A 00 In conclusion ifA12 E R with A1 A2 the trivial solution is unstable Conversely ifA12 R then A2 A1 with Mll 1 so that all solutions are bounded for all t E R Now suppose that Al A2 E R There exists a solution zt such that zt7r Alztl Either zt 0 for all t E R or zt 0 has in nitely many solutions with two consecutive zeros 21 22 satisfying 0 S 22721 S 7L In the rst case 17r A1z0 and A1i0 so that amp z7r Since zt solves equation 2 33 upon dividing by z and integrating by parts one gets that Z 013 A 048 ds 0i 41 Todd Kapitula This is a contradiction as at 2 0 In the second case suppose that 1t gt 0 for t E 21 22 Let 15 2 ter aiz For any t1 t2 6 2122 the hypothesis on at implies that 4 W 2quot 1 2 1 2 39t739t 7 2 asd32 Mag 7 lisld32 7 1sdsl M W 0 1 958 900 1 900 1 900 By the Mean Value Theorem there exists speci c t1 t2 6 21 22 such that m2 7 no 9W1 9W2 22 7 C 10 i 121 5 7 21 7 Since 121 122 0 this yields 7 22 7 21 7 1 1 4 1t2 7 1t1 7 15 Ci Zlz2 7 C 7 15 lt67 21 22 7 C gt 15Z2 7 21 Thus Z asd3gti2 7r 0 22 which yields a contradiction Hence ALA R Remark 251 Equation 234 is by no means necessary If one sets at E a3 with a0 6 RJF then equation 234 becomes a0 S 27r However in this case it is known that the triVial solution is stable for any value of a0 Note that the Floquet multipliers in this case are eiia and are realvalued and equal to unity for a0 2E Z 6 N Remark 252 The restriction on at given in equation 234 can be relaxed to asd820 lasld3 0 0 7r 6 Theorem 11182 The following general result is useful in many applications eg see 6 Chapter 1118 and 14 Lemma 253 Let t represent the principal fundamental matrix solution to equation 233 Ifl trace 7rl lt 2 then 1 0 is stable Whereas ifl trace 7rl gt 2 the solution 1 0 is unstable Proof The Floquet multipliers satisfy equation 235 ie det 7r Ang 1 furthermore by Lemma 245 trace 7r A1 A2 This yields that A1 trace 7r i trace 397r2 7 4 If trace 397r lt 2 then A1 A2 with M1 l for j 12 hence 1 0 is stable lf trace 397r gt 2 then ALA E R with A1 gt 1 which implies that 1 0 is unstable D If one denotes the principal fundamental matrix solution Via t yt then trace 1gt7r 100 yd hence one can paraphrase Lemma 253 to say that if 100 QM gt 2 236 then the triVial solution is unstable ODE Class Notes 42 3 THE MANIFOLD THEOREMS Consider the two systems 5 Am 31 and it Ax Mm 32 Where As a consequence of Theorem 229 the behavior of the flow associated With equation 31 is completely understood The stability results in Corollary 2 and Corollary 238 state that the solution behavior for these systems is asymptotically equivalent if 06A g With the additional condition that either 05A g or a A a What if the second addition is not the case The first goal herein is to show that as long as a A g then the flow associated With equation 32 is qualitatively similar to that for equation 31 In particular this Will imply that if a A f g then the solution x 0 to equation 32 is unstable As seen in the discussion leading to equation 43 it can be assumed that A diagAs Aquot Where As quot 6 R Sgt Xnsgtn With ms nu n and 0As quot 05gt A5 De ne the projection operators Hsu by HS diagus 0 Hu diagw in Where 115 E RWSWL VWSWL A Note that the projection operators satisfy the properties HsHu HuHs 0 1 1 H5 Us H i 33 furthermore HweA eA Hw 34 As a consequence of Theorem 229 one has that there exists a Ca E R such that leA Hsl g Ce m t2 0 leA Hul g Ce tS 0 35 Let 60 E R be such that for 0 lt e S 60 1 1 Clt77gt6lt1 3 6 and let 61 E R be such that for S 61 S solmll Since x is smooth and satis es the estimate 0lml2 for each given 77 E R sufficiently small there exists a 6 E R such that if lmll 12 S 6quot en W932 7 931lS 77l932 9ills Let 62 E R be such that if lmll lmgl S 62 then lrm2 T1lS 0l l21l 3 7 X I 0007ooRquot7 WM 1 suplmW 20 and for 60 I min6162 D I E X S 60 Now let mo 6 R be given so that H510 mo furthermore suppose that W S g 1 c 5 g 0 38 43 Todd Kapitula Note that for equation 31 the resulting solution satis es mm g Ce mlmol g 1 0 so 60 309 The second inequality follows from equation 38 For such an mo de ne the mapping T X gt gt X by t 00 Ty w eAquot5Hsrys d3 7 eAquot5Hurys c130 3010 0 t The last integral is wellde ned as a consequence of equation 35 If y E D then one has that t 00 M Ce mlmol CeoHyH 6W as CeoHyH em as O t 1 1 g Clmol C lt7 7gt 5060 a B As a consequence of equation 39 one then has that T D gt gt Di Upon using equation 37 one further has that for 311312 6 D l l T 7 T lt C 7 7 7 i H 12 M a B 0llll2 ylu Hence by equation 36 one has that T D gt gt D is a contraction map so that by Theorem 234 the mapping has a unique xed point ys E D Differentiating with respect to t and using equation 33 yields that ys is a bounded solution to equa tion 32 with the initial condition 00 3150 to 70 e ASHuryss dsi Thus for each bounded solution to equation 31 there exists a corresponding unique bounded solution to equation 32 Note that 00 Hsys0 10 Huys07 e Asl luryssds 0 hence there exists an hs Es gt gt E given by 00 We 2 7 eiAsnurltysltsgtgtdsy such that for the initial condition y0 0 hsmo one has a bounded solution to equation 32 It is a nontrivial exercise to show that if x is CT for some 7 E N then h5 is CT Now let 0 lt i lt a be given and It is clear that WW S Klle S Kea lml2v so that for the xed point ys to equation 310 one has the estimate em ysw S 067a7at mo CKO e7a7at7s62as yss 2613CK e at7s62as ys8 2ds L If one de nes the norm llmllw 1 supemlmwy 20 then from the above one gets that l l WltC OK 7 2 HysH wow aw Hyus ODE Class Notes 44 Hence if lt Clmol 0lmol2 which is possible since lhsmol 60060 and lmol C60 one has that S Clmol This implies that WSW S Clmole 311 Upon using equation 33 and equation 34 one further has that 00 Huyst 7 eAquot5Huryssds r so that by using equation 311 one can conclude that gymnast 0 Thus the bounded solutions decay exponentially fast as t A 00 furthermore they approach E5 in the limit Remark 31 There is an analogous result for t E R in particular the bounded solutions for t S 0 decay exponentially fast and approach E as t A 700 For equation 31 one has the existence of invariant subspaces on which the behavior of the flow is completely characterized see Theorem 229 One cannot expect invariant subspaces for the nonlinear system of equation 32 however perhaps one can expect invariant surfaces which are realized as a smooth deformation of a subspace De nition 32 A space X is a topological manifold of dimension k if each point z E X has a neighborhood homeomorphic to the unit ball in Rk In particular the graphs of smooth functions are manifolds Armed with this definition and the above discussion we are now able to state the manifold theorems for equation 32 The proofs of these theorems in the case that aCA g as well as the implications of the existence of the centermanifold WC will be given at a later time De nition 33 Let N be a given small neighborhood of m 0 The stable manifold W5 is W5 2 mo 6 N 45410 E NVt 2 0 and 45410 A 0 exponentially fast as t A 00 The unstable manifold W is W 2 mo 6 N 45410 E NVt S 0 and 45410 A 0 exponentially fast as t A 700 The center manifold WC is invariant relative to N ie if mo 6 WC then 45410 6 WC N N for all t E R Furthermore WC N W5 WC N W As already stated as a consequence of Theorem 229 one knows that the above manifolds exist for linear systems furthermore the manifolds in this case are actually linear subspaces The below results show that the inclusion of the nonlinear term 39r39m only serves to bend these linear subspaces into smooth surfaces which are tangent to the subspace at the critical point Theorem 34 Stable manifold theorem There is a neighborhood N of m 0 and a CT 1 function h5 z N WES gt gt EC 69 E such that W5 graphhs Theorem 35 Unstable manifold theorem There is a neighborhood N of m 0 and a CT 1 function h z N N E gt gt ES 69 EC such that W graphh Theorem 36 Center manifold theorem There is a neighborhood N of m 0 and a CT 1 function hC N N EC gt gt E 69 Es such that graphh is a WC Remark 37 One further has that a dimWsgtCgt dimEsgtCgt b The manifolds are invariant ie if mo 6 Wsgtcgt then 45410 E WsgtcgtLl for all t E R c WSgtCgtLl is tangent to ESgtCgtu at m 0 d The dynamical behavior on W5 and W is determined solely by the linear behavior e WC is not unique For example consider the system z39z 97y 45 Todd Kapitula 4 STABILITY ANALYSIS THE DIRECT METHOD Consider the autonomous systems 9391 f 17 41 where f E C2R is such that f0 0 The stability results presented in Section 3 rely upon a spectral analysis of Df0 Even in the case of stability no indication is given as to how close the initial data must be to the equilibrium solution in order to guarantee stability Furthermore the results contained therein leave open the question of stability in the case that 06A g In this section we will approach the stability question from a different perspective 41 The w limit set Before the dynamics associated with equation 41 can be carefully studied a mathematical description of the associated longtime asymptotics is necessary In particular a meaningful way to describe what it means for a timedependent solution to be stable is necessary The goal of this subsection is to develop this technology De nition 41 Let the unique solution to equation 41 be denoted by One says that 415 is the ow de ned by the vector eld Restating Theorem 116 Theorem 117 and Lemma 118 in terms of the ow yields Lemma 42 The How associated with equation 41 satisiies 3 M91 91 b stm tl sml sl tml c tl etml 9 Furthermore the How is as smooth as Lemma 43 If 1541 1 for all t 2 0 then 1 is a critical point If Tm m and 1511 y m for all t E 0 T then is a periodic orbit With period T In addition to talking about the ow associated with a single starting value one can discuss the ow of sets De nition 44 The ow of a set K C R is given by MK U 1511 16K Example If T is a periodic orbit then 151T F De nition 45 Let p E R be given The positive orbit 7 p is given by 7 I U MP tgo and the negative orbit 7 p is given by we r U we tgo The orbit 7W is given by 7P 1 HP U m We are now in position to describe the longtime asymptotics of the ow ODE Class Notes 46 De nition 46 The wlimit set is given by MP I H Mm I Q T 720 and the alimit set is given by MP I H MA I t T 750 Remark 47 It is an exercise to show that M y 6 R I mm A y as it A 00 aw y 6 R I MW A y as t1 A 700 The following result completely characterizes the properties of the wlimit set in the case of bounded solutions Lemma 48 Suppose that 7p 7 is bounded Then wp ap is a closed nonempty connected invariant set Proof The proof will only be given for wp as that for ap is similar Let K C R be a compact set such that 7 p C K By supposition for each 739 2 0 MW I Q T C K so that tp t 2 739 is compact In addition tp t 2 739 is connected for each 739 2 0 Finally for each 7392 gt 7391 9MP 1 t2 72 C zP I t2 71 Hence wp is the intersection of a nested family of compact connected sets so by the Bolzano Weierstrass theorem wp is nonempty compact connected set Now let us show that wp is invariant Let y E wp be given By Remark 47 there exists an increasing sequence tn with tn A 00 such that 415 p A y as n A 00 By Lemma 42 tntP tl zn Pl7 so by continuity one gets that thaw 4549 Since tn t A 00 as n A 00 for any xed t E R one then has that n13 WM 6 MP hence 1513 6 wp El Critical points and periodic orbits correspond to invariant sets What other type of orbits are to be found in wp Two examples are De nition 49 Consider equation 41 where fp0 0 and fpi 0 for p p A homoclinic orbit satis es nihinoo 410 poi A heteroclinic orbit satis es LEE 410 pi 47 Todd Kapitula 42 L yapunov functions The goal is to determine the stability of the critical point through the use of generalized energy functions De nition 410 The C1 function V R gt gt R is positive de nite if V0 0 and Vm gt 0 for m f 0 The function is negative de nite if 7Vm is positive definite lf Vm is positive de nite and if V is C3 then one has VVm 0 With 0D2V0 a D2V0i Furthermore the level set Vm 6 has a 77nice77 component surrounding m 0 for e gt 0 suf ciently small For example consider Vm 13 1 93 d3 42 Where 90 0 and ygy gt 0 for all y f 0 Since ygy gt 0 implies that fowl 98 d3 gt 0 this choice of V is a positive de nite functioni Along trajectories one has that v VV9 939 VV9 at lVVml lfml cost9i For example for the planar system i1 127 i2 g117 the function given in equation 42 satis es 0 ie it is constant along trajectories Since VVm is the outward pointing normal to the level set Vm e a minimal condition for the set bounded by Vm e to be invariant is that VVm S 0 as under this constraint the vector eld is either parallel or points into the set If V gt 0 then the vector eld points out of the set This observation leads to the following result Theorem 411 Lyapunov s Stability Theorems Suppose that V R gt gt R is positive de nite Consider equation 41 If a VVm S 0 then m 0 is stable b VVm lt 0 then m 0 is asymptotically stable 0 VVm gt 0 then m 0 is unstable Proof a For each 7 gt 0 set B0 2 m E R z lt T Let Q be the region containing the origin such that Vm is positive de nite on Q With VVm S 0 Since Vm is positive definite there is an 7 0 gt 0 such that B00 C 9 Let mo 6 B00 be given and let the solution emanating from we be denoted by By Theorem 1 16 and Theorem 120 there is a 0 lt 610 S 00 such that 610 is maximal and exists for all t E 0 moi By hypothesis and the Fundamental Theorem of Calculus z dV 2 WW 7 Wm 0 E as 0 VV s f s d3 g 0 hence S Vmo for all t E 0 zoi As a consequence of Theorem 1 16 f 0 for all t Since Vm is positive de nite one can then conclude that 0 lt S Vmoi Let 6 gt 0 be given With 0 lt e S 7 0 and set 5516Rn eSlmlSro Since Vm is continuous and Se is closed there exists a 0 I 39 V i lt M Eng 91 The lefthand inequality arises since Vm is positive de nite Since V0 0 there is a 0 lt 6 lt M such that if lmol lt 6 then Vmo lt it Thus if lmol lt 6 0 lt lt M for all t E 06moi By the definition of M ODE Class Notes 48 this implies that 55 so that lt e for all t E 0 mol Hence 610 00 and the solution is sta lei b Since 1 0 is stable it must now be shown that limtnsr00 01 Since Vm is positive de nite it is enough to show that limtnsr00 01 Suppose that there exists an 0 lt 77 lt 7 0 such that 2 77 for all t 2 01 Since Vm is continuous there is a 6 gt 0 such that if lt 6 then Vm lt 771 Since 2 77 it must be true that 2 6 for all t 2 01 Set 5516Rn 6 lml 70 and consider V I VVm on 551 By hypothesis V gt 0 is continuous on 55 so that there is a 0lt771611rV Since 0 55 and since 6 55 for all t 2 0 one has that 470150 WWW 2 M From the Fundamental Theorem of Calculus one then gets that W220 vow vlt ltsgtgt as 2 7 hence S Vmo 7 Mt For t gt Vmo77 one then has that lt 0 which is a contradiction c By the Fundamental Theorem of Calculus one has that for each t2 gt t1 2 0 mm vlt lttlgtgt 2 vvlt ltsgtgt 7718 as gt 0 hence is a strictly increasing function Let T gt 0 be the rst time that 7 0 and if no such T exists set T 001 As in the proof of b 2 5 for all t E 0T hence on the set 55 one has VV t 2 77 and consequently 2 Vmo Mt Since Vm S M for S 7 0 this yields T lt 001 1 Remark 412 The statement of part Theorem 4111c can be weakened in the following manner 4 Exer cise 1138 Suppose that Vm R gt gt R is C1 and satis es a W 0 b VVm gt 0 c Vm takes positive values in each suf ciently small neighborhood of m 01 Then 1 0 is unstable For an example consider equation 41 under the assumptions that 0A C R and that each A E 0A is semisimpler As previously discussed the second assumption is generic There then exists a nonsingular ma trix P E Rn such that PilAP A where A I diag1 1 1 1 An Upon setting 1 Py equation 41 becomes the system 739 Ag 9a 9y I P l Py 43 Note that 007 for y suf ciently close to the origin Furthermore since P is nonsingular any stability statements made regarding equation 43 immediately apply to equation 41 Now de ne the positive de nite function 1 n Vy1 71 One has that for equation 43 71 71 71 49 Todd Kapitula For each 6 gt 0 there is a 6 gt 0 such that if lt 6 then lt This follows from the fact that Suppose that 0A 05A hence there exists a M E R4r such that M S 7M lt 0 for all ii If e lt MQ then one has that n M W Emmy lt 75wa lt 0 so by Theorem 411 y 0 is asymptotically stable Similarly it can be shown that if 0A a A then y 0 is unstable as in this case Vt gt 0 for lt 6 Remark 413 It is an exercise for the student to show that a saddle is unstable The proof requires an application of Remark 412 The result of Theorem 411 is local in the sense that a de nitive statement can be made only in a suf ciently small neighborhood of a critical point The next result gives one possible way to make the result more global furthermore it precisely locates the wlimit setsi Theorem 414 Invariance principle Consider equation 41 Let V R gt gt R be positive de nite and for each k E R4r set Uk 2 m C R Vm lt k Suppose that VVm S 0 on Uk Set S I m 6 Uk VVm 0 For each mo 6 Uk one has that wmo C S In particular if0 C S is the largest invariant set in Uk then m 0 is asymptotically stable Proof The proof requires the material presented in Section 41 Since VVm S 0 on Uk one has that the set Uk is invariant under the flow furthermore V tmo S Vq 5 10 S Vmo for all s lt t E Ri For a given mo 6 Uk let p E wmoi The existence of such a point is guaranteed by Lemma 48 By De nition 46 one further has that Vp S V tmo for any t E R3 Let tn C R4r be a monotone increasing sequence with tn 7gt 00 as n 7gt 00 suc that 41510 7gt p as n 7gt 0oi By continuity one has that Vq tn 10 7gt Vp as n 7gt 00 The continuous dependence of solutions on initial data see Theorem 1 17 implies that for n E N suf ciently large and t E R suf ciently small one has that labia mo 7 tpl is small Consequently by continuity one has lVq ttn 10 7 V tpl is small for n E N suf ciently large and t E R suf ciently small Suppose that 11 5 so that lt 0 One then has that V 1P lt WP lt WNW 0 lttltlt1 so by continuity one has that for n E N suf ciently large V tnt0 lt V zn0 lt V tn7r0A Continuity then yields that V tntmo lt Vp for n E N suf ciently large and 0 lt t lt 1 suf ciently small This is a contradiction hence one must have 11 E i The stability of m 0 follows immediately from Theorem 411 Since wmo C 5 one has that distq tmoS 7gt 0 as t 7 00 lf 0 C S is the largest invariant set in Uk then one gets that m 0 is asymptotically stable 1 For an example consider van der Pol7s equation i2p1712iz0 Mgt0i ln Lienard form it is written as 1391 72le1 7 12 1392 711 Consider the positive de nite function Vm One has that VV91 f91 ZWW 7 Ii3 ODE Class Notes 50 If one sets UgQ m E R2 Vm lt 32 then VVm g 0 for m E UgQl Set SmEU32 2110 Now 1391 12 f 0 except at 00 Hence the origin is the largest invariant set in 5 so by Theorem 4 14 1 0 is asymptotically stable 421 Example Hamiltonian systems Hamiltonls equations of motion are given by 76H 17 6H apiy Pzi aqiv 4 z391mn 44 where H Hp q E C2R2nl Note that upon setting 1 I qpT equation 44 can be written as 5 JVH1 I z lt g 35 45 For example if one considers i 0 46 then by setting 417 2 one has the Hamiltonian Hltpqgt p2Fltqgt 1 qfltsgtds 47 In equation 47 one has the physical interpretation that 1022 is the kinetic energy and Fq is the potential energy One has that n n 8H 8H H it i 0 1 64139 Z 610139 1 hence one can use the Hamiltonian as a Lyapunov function Without loss of generality one can assume that H0 0 If VH0 0 ie if m 0 is a critical point for equation 45 and if H is positive de nite ilel 0D2H0 a D2H0 then by Theorem 411 the origin is stable The conclusion still follows if H is negative de nite ilel 0D2H0 05D2H0 if instead of taking H as the Lyapunov function one takes iHl Now assume that 1 n HQ 4 5 10 q i1 where 0 and V 0 0 Upon a change of coordinates one can write 1 n 154 i Edit 0l4l3l i1 In this new coordinate system the Hamiltonian equations are 4139 Piv 10239 ain39 0l4l2l The linearization about the critical point yields the eigenvalues A ideall Note that these eigenvalues are generally semisimpler If 45 is positive de nite ilel i ai E R for each i then it is easy to show that H is positive de nite hence the origin is stable lf ai E R for some but not all 239 ie if 45 is not positive de nite at q 0 then the origin is a saddle point and by Remark 413 is consequently unstablel 51 Todd Kapitula For example again consider equation 46 With the associated Hamiltonian given in equation 47 Assume that FO 0 and note that F 0 f 0i From the above discussion one has that if f 0 lt 0 then the origin is unstable Whereas if fO gt 0 the origin is stable Thus minima of F correspond to stable critical points While maxima correspond to unstable critical points This corresponds to the physical intuition that minimum points of the potential energy are stable While maximum points are unsta er ODE Class Notes 52 5 PERIODIC SOLUTIONS Again consider the autonomous system 939 mg 51 where f R gt gt R is smooth Recall that from Lemma 48 it is known that for bounded trajectories the wlimit set is compact7 connected7 and invariant In Section 4 some conditions were given which guaranteed that this set was the critical point x 0 In this section we will be concerned with the existence and nonexistence of periodic solutions Most of the results given herein will be applicable only in the case that n 2 as the topology of the plane allows one to make more de nitive statements regarding the wlimit set In fact7 unless otherwise stated it will henceforth be assumed that n 2 51 NoneXstence Bendixson39s criterion In this subsection we will give a criteria which guarantees that no solutions exist to equation 51 Before doing so7 however7 one needs to be reacquainted with Greenls Theorem and the Divergence Theorem This in turn requires the following characterization of closed continuous curves 7 C R2 Theorem 51 Jordan Curve Theorem A simple closed continuous curve 7 C R2 divides the plane into two connected components One is bounded and is called the interior of 7 and the other is unbounded and called the exterior of the 7 Each component has 7 as its boundary De nition 52 If 7 C R2 satis es the Jordan Curve Theorem7 set int7 to be the interior of 7 and ext 7 to be the exterior of 7 Theorem 53 Let f fl7 f2 R2 gt gt R2 be smooth Let 7 C R2 be a Jordan curve bounding a domain D One has the following results a Green s theorem ff dR 8751162 7 8752161 dA y D b Divergence theorem ffldlg 7 fgdzl VfdA y D Where V 39 f 1 6021 f1 amfa If in Theorem 53 one thinks of 7 as representing a invariant Jordan curve7 eg7 a periodic orbit7 for equation 517 then one can interpret Green s Theorem and the Divergence Theorem as giving necessary conditions on the vector eld for the existence of 7 In particular Theorem 54 Bendixson s criterion Consider equation 51 When n 2 Let Q C R2 be a simply connected region lfV f 0 for all m E Q then the system has no invariant Jordan curves contained in 9 Proof Suppose that there is an invariant Jordan curve 7 C Q Parameterize the curve so that it is traversed once in the counterclockwise direction for 0 S t S 1 Set D I int7 Since the curve is invariant one has that the vector eld f is tangent at all points consequently7 one can say without loss of generality that ii along the curve One then has that 1 d d f1d12 f2d110 f1 f2 dt0 By Theorem 53b this then implies that DvfdAo which contradicts the assumption that V f never changes sign Hence7 7 does not exist D 53 Todd Kapitula For the rst example recall that in Section 42 van der Pol7s equation in Lienard form was given by 1391 72le1 7 12 1392 711 Further recall that if mt 6 Us I m E R2 z I 1 lt 3 for any value oft then mt 6 Us for all t E R with mt A 0 as t A 00 Now V f 7217 Upon setting UizmeR2 11gt1 UlmeR2zzllt71 an application of Theorem 54 then yields that any periodic solution 7 must satisfy 7 C lR2Us with 7 gZ U and either or both 7 U f g For the second example consider 3395 MINE 41 0 Suppose that 101 gt 0 damping For the system i1 127 i2 7411 P11I2 one has that V f 71011 lt 0 Hence by Theorem 54 no periodic solution exists In fact if q0 0 and 141 gt 0 for all I f 0 then by using the appropriate Lyapunov function one has that for any 10 zt A 0 as t A 00 52 Existence Poincare BendIXSOn Theorem Now that one has criteria for which no periodic solutions exist it is time to develop conditions under which one can guarantee the existence of such solutions The notation and ideas presented in Section 41 will be used extensively here Again consider equation 51 in the case that n 2 Let 7 I 45110 0 S t S T be a periodic orbit with minimal period T Let 1 E R2 be chosen so that v fp 0 The vector v is said to be transversal to 7 at the point p For 6 gt 0 set L5216R2 m pav lal g 6 L5 is said to be a transversal section to 7 at p Since fp 0 e gt 0 can be chosen suf ciently small so that L5 7 p and that all orbits crossing L5 do so in the same direction Lemma 55 There is a 6 gt 0 such that ifmo 6 L5 then there is a continuous Tmo gt 0 With limlohp Tmo T such that Tzoo 6 Le Proof Since f is smooth the ow depends smoothly on 1 Applying the lmplicit Function Theorem to Ct m I v note that GTp 0 yields the result El Remark 56 If f R gt gt R the transversal section is de ned by n71 21 E R z m p Zaivi lail g e where v1 vn1 is a linearly independent set which satis es vi 0 for all 239 Lemma 55 allows one to de ne a smooth map near a periodic orbit De nition 57 The return time map is given by Tm and the Poincare map is given by 111 T1m Lemma 55 allows one to understand the dynamics near a periodic orbit via a study of a map versus the study of the full ow The advantage to this approach is that the dimensionality is reduced by one However even in the case that n 2 this does not necessarily imply that the problem is easy eg see 7 Chapter However one recovers periodic orbits quite easily as xed points of H ODE Class Notes 54 Lemma 58 Let mo 6 E Where E is de ned in Remark 56 Hmo mo if and only if 45410 is a periodic orbit With minimal period Tmo Proof By de nition Hmo mo if and only if T10mo 0 for some Tmo gt 0 The result now follows from Theorem 16 El Consider equation 51 when n 2 Suppose that there is a section L C R2 such that for the orbit 415 p one can de ne a Poincare map H L gt gt L Without loss of generality it can be assumed that L is a subset of the zlaxis so that the map can be represented by x10 gt gt 1111 0 By Theorem 116 and smoothness one has that the map is a diffeomorphism For a given point 11 0 E L one has three possibilities for the map a 1111 11 b 1111 gt 11 or c 1111 lt 11 Case a implies that 110 is contained in a periodic orbit The uniqueness of solutions implies in cases b and c that the sequence Hj will either be monotone increasing case or decreasing case for all j E N If one assumes that 7p is bounded then one has that the sequence is bounded and hence has a limit point The following theorem allows one to characterize the orbits associated with these limit points Theorem 59 Poincar Bendixson Theorem Consider equation 51 under the assumptions that n 2 and that there exist only a nite number of critical points Suppose that for some p E R2 7 p is bounded Then one of the following holds a wp is a critical point b wp is a periodic orbit c wp is the union of nitely many critical points and perhaps a countably in nite set of connecting rbits In cases b and c wp satis es the Jordan Curve Theorem Proof Eg see 17 Chapter 43 The basic idea is to carefully study the properties of the relevant Poincare map El Remark 510 One has that a if 7 p is bounded then there is a similar result for 11 b case c allows the existence of heteroclinic and homoclinic orbits An easy consequence of Theorem 59 which is especially useful in applications is the following Corollary 511 If there is a positively invariant region Q Which contains no critical points then 9 contains at least one periodic orbit If 7 C R2 is a periodic orbit one can characterize its stability in the following manner De nition 512 A periodic orbit 7 C R2 is a limit cycle if there is a pl 6 int7 and pg 6 ext such that either Wm mp2 v or am aw m 521 Examples Example Consider i1 611 7 12 31 2111 i2 11 512 31 2112 In polar coordinates 11 I 7 cos 9 12 I 7 sin 9 the system is T 2 cos2 t939r2 1 If B gt 0 then gt 0 so that all solutions are unbounded as t A 00 hence there exist no periodic orbits As t A 700 all trajectories approach the origin 55 Todd Kapitula Suppose that B lt 0 For 0 lt 6 ltlt 1 consider the annulus D52 049 z 7366 ltT2 lt72i6 Since T 2r2 lt 39f lt T 37quot2 D5 is negatively invariant ie the vector eld points out on the boundary of Del Thus if p 6 D5 7 p satis es the hypotheses of the Poincar Bendixson theoremi Since D5 contains no critical points 11 is a periodic orbiti Unfortunately the Poincar Bendixson theorem says nothing about the number of periodic orbits in Del However since D5 is an annulus which is not simply connected by appropriately modifying the proof to the Bendixson criterion see 15 pl 262 one cay say the following Lemma 513 Consider i fm Where f R2 gt gt R2 is smooth Suppose that there is an annular region Q C R2 such that V f 0 for all m E 9 Then there exists at most one periodic orbit 7 C 9 Example I conti A routine calculation shows that V 25 22 cos2 t939r2 so that V gt 0 for 7 2 gt 7641 Thus V gt 0 for all m 6 D5 so that by Lemma 5113 there is only one periodic solution contained in Del Also lt 0 for 7 2 lt 753 and gt 0 for 7 2 gt 762 implies that neither of these regions contains a periodic orbiti Hence the periodic orbit contained in D5 is unique Remark 514 In general an invariant region may contain more than one periodic orbiti Consider 407 no 72 911 The annulus DT9 ltr2ltg is positively invariant and contains the two periodic orbits 7 1 2 Example II Recall that Van der Pol7s equation in Lienard form is given by 1391 72le17 12 1392 711 Suppose that M lt 0 It can be shown that the set Snag I z E R2 z I 12 lt 3 is negatively invariant furthermore if 10 E Snag then zt 7gt 0 as t 7gt 700 Furthermore it has been seen that any periodic solution must intersect either or both of 11 i1i Let us now show that such a periodic solution exists in a particular limiti Set 6 I 1AM i2 2 612 and rewrite the equations upon removing the hat as ail 21117 12 1392 7611 This is a singular system for the vector eld is not smooth as e 7 01 Set 8 I te so that the equations now become 2 dds 11 21117 12 12 762111 Let us rst study this new system in the case 6 0 The lines 12 C are invariant and on these lines the ODE is given by 11 21117 12 Thus one can construct an invariant Jordan curve say 7 composed of critical points and heteroclinic orbitsi It can be shown that given 6 gt 0 there is an 60 gt 0 and an open set U lying within a distance 6 of 7 such that U is positively invariant for 0 lt e lt so This set U will contain no critical points so by the Poincare Bendixson theorem there will exist a periodic orbit in U1 The proof is by picture and uses the fact that 12 76211 for e gt 0 The resulting periodic solution is an example of a relaxationoscillation a periodic solution operating on different time scales See 17 Chapter 1213 for further details ODE Class Notes 56 Example III Let 71 and 72 be two invariant Jordan curves and suppose that 71 C int72 Set D int72 ext71 Suppose that there exist no critical points or periodic orbits in D Let mo 6 D be given Since D is bounded and invariant wmo C D Since D contains no critical points or periodic orbits by the Poincar Bendixson theorem one has that either wmo 71 or wmo 72 Without loss of generality suppose that wmo 71 Let us now show that for p E D wp 71 Suppose that there is an ml 6 D such that wml 72 Let E be the line containing mo and ml which transversely intersects 71 and 72 Pick a point 12 E E which is between 0 and 11 By the uniqueness of solutions 7 m2 is trapped between 74r mo and 74F 11 which implies that 7 m2 is uniformly bounded away from both 71 and 72 By the Poincar Bendixson theorem 112 is either a periodic orbit or contains critical points Since 112 C D and 112 m 71 U 72 this implies that D itself contains either a periodic orbit or critical points This is a contradiction thus wml 71 5 3 Index theory Given a periodic orbit 7 C R2 it is natural to inquire as to what types of orbits reside in int7 In particular does int7 necessarily contain critical points If so is the nature of the ow near the critical point necessarily proscribed An application of the following theorem yields an answer to the rst question Theorem 515 Brouwerls xed point theorem Let U C R be homeomorphic to a closed ball in R Let g U gt7 R be continuous and satisfy 98U C U Then 9 has at least one xed point in U ie there is at least one 1 E U such that 91quot 1 Theorem 516 Let 7 C R2 be an invariant Jordan curve Then int7 contains at least one critical point Proof If 7 C R2 is a Jordan curve then int7 is homeomorphic to a closed ball in R2 Since the ow is continuous and satis es 415 int int 7 one can attempt to apply Theorem 515 to deduce the existence of an equilibrium point For a given 11 E int7 one has that 454p E int7 for all t 2 0 in particular this implies that for a given t1 gt 0 one has that 451 int7 gt7gt int7 Thus by Theorem 515 there is a pl 6 int7 such that 4511 p1 p1 Choose a decreasing sequence tn with limnn00 tn 0 and get the corresponding sequence pn with 41511n 7 pn Without loss of generality suppose that limnn00 p7 p For each t E R and any n E N there is a kn E Z such that kntn S t lt kn ltn so that 0 S t7 kntn lt tn Hence given 6 gt 0 there is a 6 gt 0 such that if tn lt 6 then Wkkntn p7 7 pnl lt 63 By using the smoothness of the ow one has that there is an N1 2 1 such that if n gt N1 then l tpn 7 tpl lt 63 Finally there is an N2 2 1 such that lpn 7 p l lt 63 if n 2 N2 Now using the properties of the ow detailed in Lemma 42 one has that zPn z 7z z 7kz 24th Thus for N 2 maxN1 N2 one has that WW 7 W S WW 7 Mum WM 7 ml m 7 lt 6 This implies that qb p p for all t E R which means that p is a critical point El As seen in the next example the result of Theorem 516 can be used to show that a system possesses no invariant Jordan curves Example Consider the system 1391 17 1 1392 1112 Since the system has no critical points by Theorem 516 there exist no invariant Jordan curves Note that Bendixsonls criterion does not yield any information as V 11 Now that it is known that periodic orbits must contain critical points in the interior the answer to the question regarding the nature of the critical points can now be pursued Rewrite equation 51 as the nonautonomous scalar equation 6112 7 f2117I2 01711 7 f111712 57 Todd Kapitula mm 16111712 9 is the angle the vector eld makes with the positive zlaxis Let 7 C R2 be a positively oriented Jordan curve which does not contain any critical points of f The index of f with respect to 7 is given by 397 7i flde defl m 7 2W ewe 2W f12f22 52gt tant9 jf 7 represents the number of multiples of 27f the angle that f makes with the positive zlaxis changes as 7 is traversed once One has that jf 7 varies continuously with any continuous deformation of 7 which does not lead to encounters with critical points As a consequence of 157 Theorem 3121Corollary 3121 one has that Lemma 517 Let 7 C R2 be a positively oriented Jordan curve Which does not contain any critical points of f Then a if int7 does not contain any critical points then jf 7 0 b if71 and 72 are Jordan curves With 71 C int 72 and ifthere are no critical points in int 72 ext 71 the if 71 J39f7239 As a consequence of Lemma 5177 one can de ne the index of a critical point we in the following manner Let 7 be a Jordan curve such that mo 6 int7 and that int7 contains no other critical points off Under this scenario set J39f9lo 117 1f 7 encloses a nite number of critical points7 then a proper application of Lemma 517 see 157 Theo rem 3122 yields the following Lemma 518 Let 7 C R2 be a positively oriented Jordan curve Whose interior contains the critical points ml mn en MW Zi m 161 It is now time to understand the manner in which one can compute jf mo7 where we is an isolated critical point Let f and g be two vector elds such that fmo 910 0 Further assume that 91 is a continuous deformation of For a given 6 gt 0 suf ciently small one has that there is a 6 gt 0 such that for 7 2 8B 06 one has 7 lt 6 Since f 0 on 77 by making 6 suf ciently small one can guarantee that f and g roughly point in the same direction all along 7 By de nition this necessarily implies that jf930 J39g93o 53 In other words7 the index is unchanged relative to small perturbations of the vector eld This observation leads to the following result see 157 Theorem 3125 Lemma 519 Consider it Ax 39rm7 Where One has that jA10 jA1r10 As a consequence of Lemma 5197 in order to compute the index of a critical point it is suf cient to compute the index of the associated linearized problem Using the de nition in equation 52 yields that in general7 detA 11 dzg 7 12 dzl 11410 7 f lt54 2 Y allzl a12r2 2111 a2212 ODE Class Notes 58 Henceforth assume that detA 0 Now7 it can be shown that the index is invariant under a nonsingular linear transformation hence7 when computing jA10 it is sufficient to consider those A which have the Jordanforms a 0 a1 a 7b Ar7lt0bgt Ad7lt0agt AC7ltb a Assuming that a f 0 for Ad one has that there is a continuous deformation such that either 0 39 Ad H lt alga a 62 gt7 s1gna 61 SlgTIWJF 52 Slgna a e 7b 7 Adgt gtlt b a6gt s1gnae7s1gnabElR As a consequence of the discussion leading to equation 53 one then has that jAd10 ijE0 in the case that signlz signa7 or jAd10 jACE0 Hence7 it is enough to compute the indices only in the cases of Ar and C First consider Ar Upon evaluating equation 54 over the ellipse l l 7 1112 lt7 cos t7 E sintgt 0 S t S 2W7 a and noting that the curve is positively oriented if ab gt 0 and negatively oriented if ab lt 07 one sees that 71 ablt0 10 M 1 abgt0 Now consider AC Evaluating equation 54 over the positively oriented unit circle quickly yields that jACE0 1 The following result has now been proven Lemma 520 Consider A E R2 under the condition that detA 0 One has that jA10 71 if is a saddle point otherwise jA10 1 As a consequence of Lemma 519 one has the following result concerning critical points of nonlinear systems Corollary 521 Consider T53 f 17 Where fmo 0 Assume that detDfmo 0 One has that m 7 71 mo isasaddle point Jf 0 7 1 otherwise Now suppose that 7 is a Jordan curve which is invariant under the ow It may be possible that 7 contains critical points henceforth7 it will be assumed that there exist at most nitely many The de nition of jf 7 given in equation 52 requires that no critical points be on 7 however7 this technical difficulty can be overcome 157 Remark 3121 The proof of the following result is that for 157 Theorem 3123 Theorem 522 lf7 be is a Jordan curve Which is invariant under the How then jf 7 1 By applying Corollary 521 to Theorem 522 one gets the following result Corollary 523 If 7 is an invariant Jordan curve Which encloses only one critical point then that point cannot be a saddle point Proof Suppose otherwise By Theorem 522 one has that jf 7 1 whereas by Corollary 521 one has that the index of a saddle point is 71 This is a contradiction7 as the index is invariant under continuous deformation of 7 59 Todd Kapitula Example Consider 1391 zlzg 1392 11 72zgziz The only critical point is 00 This critical point is a saddle point7 as the eigenvalues associated with the linearization are A 71 i 132 Hence7 there exists no periodic solution to the system Note that Bendixsonls criterion does not yield any information7 as V 71 31 The proof of the following nal result is left for the student Lemma 524 Consider it f Assume that for each critical point we one has that detDf 10 f 0 Let 7 be a Jordan curve Which is invariant under the How Then a int7 must contain an odd number of critical points b of the 2n 1 critical points contained in intoy7 n are saddle points 54 Periodic vector fields Now consider the following variant of equation 51 9391ft7937 ftTft7937 55 where f R X R gt gt R is smooth As an autonomous rstorder system this can be rewritten as aquot ftm i1 ie7 for y I ttT7 1391 my 9y1fy71T 5 6 Recall the discussion on Poincare maps for equation 56 leading to De nition 57 A Poincare map for periodic vector elds of equation 55 can be de ned in the following manner For E E Z set 212 tm ERX R tZT7 and identify 2 I 20 with 21 so that E is a Poincare section on the cylinder The Poincare map for equation 56 will be de ned by Py I Equivalently7 if 1010 is the solution to equation 55 with the initial data 10 mo then one has Pmo 1T mo A de nitive relationship between the solution and the iterates of the Poincare map is given in the following result Lemma 525 Set Pk 2 Po P 1 for k 12 Then Pkmo 1kT 0 for k E Z Proof Since ft T7 1 ft7 m by a standard induction argument one has that ft kT m ft7 m for each k E Z Set zt I mt kT mo Then Iz ft7 z7 20 1kT mo so by uniqueness one must have that zt mt 1kT 10 In other words7 mtkT mo mt 1kT 10 Now7 Pmo 1T mo and P2mo P o Pmo 1T 1T 10 12T 0 The rest of the proof follows from an induction argument 1 Corollary 526 The solution 1010 to equation 54 is kT periodic if and only if Pk mo mo Proof Suppose that Pk mo mo By Lemma 525 this implies that 1kT mo mo so that mtkT mo mt 1kT 10 0210 Thus7 the solution is kTperiodic The other direction follows by the de nition of the Poincare map and Lemma 525 D ODE Class Notes 60 As a consequence of Corollary 5126 one can discuss periodic orbits for the Poincare mapl De nition 527 The point p is a periodic point of period N if PNp p but P1601 p for k 11 N 71 The periodic orbit for the map is given by 11 Pp l l l PN 1pl Example For a simple example consider the damped and periodically forced harmonic oscillator modeled by i 2M1 z hcoswt where M 6 01 and hw E R4r also see 17 Example 514 If M 0 it will be assumed that w 1 ie there will be no resonant forcingl The solution to the initial value problem is given by 1t cleiw cos1 7 M2 t Cgei t sin1 7 M2 t Acoswt B sinwt where 1 7 0J2 L 4M2w2 7w22 A B 397 2Mw 7 4M2w2 17w22 Mz0 7 AM 7 Bw 17 M2 For 1 11T and T I 27rw the Poincare map is given by P10 1T ie 611 10 7 A 52 I Pltmlt0gtgt 7 cle T cos702e T sin7 A 7 701M02 17M2e Tcos7clxl7M252Me Tsin7Bw 7 where 7 I l 7 M2 T First suppose that M gt 0 The unique xed point is then given by 1 I ABwTl Furthermore it is not dif cult to show that lim P 10 1 n7gtoo hence the periodic solution is asymptotically stable Now suppose that M 0 One then has that h 17w27 A B0 01107A 0210 so that the Poincare map satis es PM 7 lt sis gtmltogtAlt 1 7cosT If an Q then the unique stable xed point is given by 1 I A0Tl Furthermore as long as w Q then P2 has a unique xed point 1 for each Z 6 N If an E Ql then there is still a unique stable xed point for P however PI 11 for some Z 6 Nl so that in this case all solutions are ZENperiodic Consider the general problem of nding xed points for Pl If there is a closed ball B C 2 such that P8B C B then as an application of Theorem 5115 one gets the existence of a xed point of P ie a point 1 E B such that P1 1 By Corollary 5126 this in turn implies the existence of a periodic orbit 1t In general not much more can be said about the Poincare mapl However this is not true in the case of scalar vector elds In particular one can use the uniqueness of solutions to determine the behavior of the sequence Pn 100U Proposition 528 Suppose in equation 55 that f R X R gt gt R Then P zo0 is a monotone sequence Proof Without loss of generality suppose that Pzo gt 10 De ne the sequence of solutions 2k 0 T gt gt R by 2160 I HEPC10 Note that 2k T Pk1 10 By the uniqueness of solutions 21t gt 20 t for all t E 0 T hence P2zo gt Pzol An induction argument yields that Pn1 10 gt P zo for all n E N 61 Todd Kapitula Remark 529 If Pzo lt 10 then the sequence is monotone decreasing An application of Proposition 528 yields a Poincar Bendixson type theorem for scalar periodic vector elds Theorem 530 Consider equation 55 in the case that f R X R gt gt R If the solution zt 10 is uniformly bounded then there exists a T periodic solution Proof By Proposition 528 the sequence P 10 is monotone By supposition this sequence is bounded hence there exists a y E R such that P zo A y as n A 00 Now by continuity one has that 1T y lim 1T P zo lim 10 Pn1zo 10 lim Pn1zo 10 y hence zt T y zt y for all t 2 0 D Given the existence of xed points for the Poincare map one de nes stability as below De nition 531 p is a stable xed point of P if for each 6 gt 0 there is a 6 gt 0 such that if lz 7 pl lt 6 then 7 pl lt e for all n E N Otherwise the xed point is unstable The xed point is asymptotically stable if it is stable and P A p as n A 00 Example Demonstrate the graphical iteration of scalar maps Theorem 532 Let P R gt gt R be a C1 map A xed point p is asymptotically stable iflPpl lt l and unstable iflP p gt 1 Proof Upon setting u 2 zip and gu I PuP Pp the map zn1 Pzn becomes un1 Noting that 90 0 one then has that without loss of generality the xed point is p 0 Since P0 0 by the Fundamental Theorem of Calculus one has that Pz Om P s d3 Given 6 gt 0 set met min P z M5max P 1 Mi l Ml l thus for S 6 one has melzl S S Upon repeated applications of the chain rule one can then show that mglrl S anIl S Menlrl Suppose that lP 0l lt 1 so that for e gt 0 suf ciently small one has that M5 lt 1 Then for 6 eMe and lt 6 one has that lt 6 so that the xed point is stable Furthermore since lim S lim 0 ngtw ngtm the xed point is asymptotically stable Now suppose that lP 0l gt 1 For 6 gt 0 suf ciently small one then has that me gt 1 so that as long as S 6 Given an 60 lt e and 10 with lzol lt 60 there is an N such that lPNzol mylzol gt 60 Since 10 is arbitrary the xed point is unstable 2 2 It is now necessary to understand how one can compute the derivative of the Poincare map at a xed point Let 7t be a Tperiodic solution such that 70 p Let the solution to equation 55 be denoted 1010 Since Pzo 1T 10 we have that PIo diNT ol 10 Upon using the chain rule and smoothness one gets that d d d a Mth The equation is linear and since d 017010110 17 ODE Class Notes 62 it has the solution d t 710210 exp f szs 10 d3 dzo 0 Evaluating at t T and 10 p yields the following result Lemma 533 Consider equation 55 in the case that f R X R gt gt R Let 7t be a T periodic solution such that 70 p Then T Pp eao a0 0 f t7t dt Note that Pp gt 0 for any xed point p As an application of Theorem 532 one has the following result Corollary 534 If a0 lt 0 then the xed point is asymptotically stable Whereas if a0 gt 0 then the xed point is unstable Example Consider equation 55 in the case that ftz 713 at7 where at T at It will be shown that for this vector eld that there is a unique asymptotically stable Tperiodic solution First suppose that Pp p for some p and let 7t be the corresponding Tperiodic solution Since a0 lg 7372 t dt lt 07 by Corollary 534 the xed point is asymptotically stable In order to show that the xed point is unique7 set 91 2 z 7 Note that at a xed point 417 ie7 gq 07 one has that gq 17 P q gt 0 Let 11 lt 12 be two xed points such that 91 0 for 11 lt z lt 12 Since gzi gt 0 for i 12 by continuity there must exist a point 13 E 117 12 such that 913 0 with gzg S 0 This is a contradiction hence7 there can exist at most one xed point It is now time to show that the assumed xed point7 which is unique7 actually exists Since at is continuous and periodic7 there exists an M gt 0 such that latl S M for all t E QT Set U Zt1 713 7 M gt 07 U Zt1 713 M lt 0 If I E U then 139 gt 07 while if z E U7 then 139 lt 0 Thus if z E U one has that Pz gt I while if z E U one has Pz lt I Since the Poincare map is continuous7 there is a point p such that Pp p 63 Todd Kapitula 6 APPLICATIONS OF CENTER MANIFOLD THEORY Again consider the autonomous system 939 aw 61 where f R X Rk gt gt R is smooth and satis es f0 0 0 Recall the statements of the manifold theorems given in Section 3 While it is not explicitly stated therein it can be shown that these manifolds vary smoothly with respect to parameters Now suppose that dimEC E where l S E S n As an application of the Center Manifold Theorem 36 there exists an invariant Z kdimensional invariant manifold so that the governing equations are of dimension Z kl This reduced set of equations will determine all of the interesting solutions near 1 0 as it is known that any solutions on W5 W will have exponential behavior as t A 00 t A 700 The manifold theorems are also important in that they implicitly tell us how to compute the ow on War Consider equation 61 written in the form yAMyyly727 7 Z BMzy2yzw7 62 where y 6 ES 69 E z 6 EC and lgj y zl 02 where 0m implies the inclusion of all terms of the form with ij k 2 mi As a consequence of the Center Manifold Theorem 36 it is known that WC is given by the graph y hC 24 where RDhc00 E9 The local ow on WC is then given by Z 3002 92016940 27M 63 Since hC z M is smooth by using the invariance property of the manifold it can be computed via a Taylor expansion 61 Reduction to scalar systems As a rst example consider the case that M E R and that 71 0 Df00 7 lt 0 0 gti 64 In this case there is a onedimensional stable manifold and a onedimensional center manifold furthermore the stable manifold is tangent to spanel and the center manifold is tangent to spanegi It can be shown via the theory of normal forms eg see 18 Chapter 19 that equation 62 can be written as 139 71 almz blzy 03 6 5 y39a2a3yb292037 A where the constants a b39 E R Henceforth ignore the 03 terms as an application of the lmplicit Function Theorem makes them irrelevanti For the truncated equation 65 the line I 0 is invariant hence it is the center manifold The ow on the center manifold is then governed by y aw awy my 66 Assuming that 122 f 0 rescale equation 66 via 8 I lbglt so that equation 66 becomes d y 042M0 3y5927 1 37 67 where 6 E ill and aj ajlbgli The analysis of equation 67 is straightforward If 12 f 0 the y critical points are given y ix 6a2wr 00M 64 ODE Class Notes This is an example of a saddlenode bifurcation 18 Chapter 201c Note that the expression makes sense if and only if by E R7 The ow on the center manifold is depicted in Figure 8 and the full ow near the b origin is depicted in Figure 9 If 12 0 the critical points are given y y i aw y 07 This is an example of a transcritical bifurcation 18 Chapter 201d The student is inVited to draw the bifurcation diagrams similar to those in Figure 8 and Figure 9 for the saddlenode bifurcation Figure 8 The bifurcation diagrams for equation 67 in the case that 12 7 O and 6 1 ugt0 ult0 x Figure 9 The ow near the origin for equation 67 in the case that 6 1 and 12 gt 0 Before considering the next example7 the following result is needed Proposition 61 Suppose that g R gt gt R is C2 in a neighborhood of m 0 and suppose that 107 12 zn 0 for all 07 12 zn in a neighborhood of the origin There exists a neighborhood M of the origin and a 91 E C1 such that 91 zlgl Proof By Taylorls theorem one has that 1 8 91 y0zgzn Eg zhzgpuwwdt 0 65 Todd Kapitula Choose M so that the line between 1 and 012 l i i InT lies in Mr This yields that a a miyrauwrn I1D9t117127m71n817 which in turn implies that 1 911 DgtzlzgiuzneldteC1Mi D 0 For the next example consider the system i 7x 12 7 y2 3 68 y 51 my 7 y Note that the critical parameter 2 present in the normal form of equation 65 has been set to zero in equation 68 At the critical point zye 0 0 0 one has that 05Df0 71 Es spanel aCDf0 0 EC spanegi Upon applying the Center Manifold Theorem 36 one knows that the center manifold is locally given by z hye h0 0 hy00 h500 0 69 and the ow on WC is given by y39 eyyhye 7y3i 610 The function hy 6 must now be determined It is clear that 006 is a critical point for any 6 E R hence h0 e E 0 so by Proposition 61 one can write hy e yh1y e As a consequence of the smoothness of the vector eld the function hl has a Taylor expansion which by equation 69 is given by h1ye ay be 02 Since WC is invariant one has that i 7 e 6 7 Byy 86 7 which yields that 7hy e hy e2 7 y2 Zay be 02ey yhy e 7 ygi Simplifying the above expression gives a 092 7 hey 03 03 which necessarily implies that In conclusion hy76 90y 02 611 Substituting the result of equation 611 into equation 6 10 yields that the ow on WC is given by y39 96 i 2y2 03 612 Set my761 6 i 2f 03 Since m00 0 and me 00 l by the lmplicit Function Theorem there exists an e 6y and yo gt 0 such that me y E 0 for all lt yer By inspection one has that 6y 2y2 03 The above example yields what is known as a pitchfork bifurcation 18 Chapter 20ileli The student is invited to draw the bifurcation diagrams for this problem similar to those in Figure 8 and Figure 9 ODE Class Notes 66 611 Example singular perturbations Consider the system 5 IMyIy 613 59 I i y 197 where M 6 01 and 0 lt 6 ltlt 1 The system arises as a model of the kinetics of enzyme reactions see 37 Example 143 and the references therein Setting 8 2 6t and 2 I z 7 y transforms equation 613 to the system 1 efzz 2 212 7zzefzz 614 where I dds and fzz I 7x z Mz 7 Upon applying the Center Manifold Theorem 36 one knows that the center manifold for equation 614 is locally given by Z MRS h070 hx070 he070 07 and the ow on WC is given by 1 efzhze 615 The function hze must now be determined As a consequence of the smoothness of the vector eld the function h has a Taylor expansion which is given by hz7 e a62 1261 012 03 Since WC is invariant one has that ah ah 7 7 7 2 7611 666 937 which eventually yields that 03 7ae2 01 717 bez 17 cz2 03 03 This necessarily implies that a07 b 1 7 61 so that one can conclude that hz7 e 12 7 17 Mex 03 616 Substituting the result of equation 616 into equation 615 yields that the ow on WC is given by 1 6 717 it 12 03 617 hence7 there is a transcritical bifurcation The argument for the irrelevance of the 04 terms is the same as for the previous example The student is invited to draw the bifurcation diagrams for this problem similar to those in Figure 8 and Figure 9 612 Example hyperbolic conservation laws A viscous conservation law is given by u fux 1117 618 67 Todd Kapitula where f R gt gt R is Cm A thorough discussion of conservation laws and their importance in applications can be found in 16 Part III The goal here is to nd travelling waves which are solutions uz 2 I z 7 st of equation 618 which satisfy the asymptotics uz 7 Z A 700 619 R 2 7 00 In the travelling frame equation 618 is written as u 7 311 u u 620 and the travelling wave will now be a steadystate solution ie a solution to 7 311 u u 621 It will be realized as a heteroclinic orbit The following assumption will be required Assumption 62 The function f satis es a for all u E R Dfu has distinct real eigenvalues A101 lt A201 lt lt Anu ie the system is strictly hyperbolic b ltVAju 39r39j lt 0 for all u E R where 39r39j is the eigenvector associated with the eigenvalue Aju ie the system is genuinely nonlinear Remark 63 If n l Assumption 62b is equivalent to specifying that is convex Furthermore it guarantees that for each s lt fuL there is a unique uRs gt uL such that there is a solution to equation 621 which satis es equation 619 Lemma 64 For each uL E R and each 1 S k S n there exists a curve of states ul m for 0 lt p lt p0 such that a travelling wave exists With speed s 8k Furthermore a ul m and 3km are CT for any 7 E N and 139 k 139 k A pgg rd L 31907 MILL b Akwl lt SW7 lt AMUL c Amum lt 8 lt Ak1uf Remark 65 Conditions b and c are known as the LaX entropy inequalities Proof Set 39 I ddz lntegrating equation 621 from 700 to 2 and using the fact that 7 UL as 2 7gt 700 yields it u fWL 3W UL Linearizing at the critical point uL yields A I DfuL 7 31 Upon noting that UA A 7 s z A E 0DfuLv it is seen that a bifurcation can occur only if s AkuL for some k ln Note that by Assump tion 62a the equation describing the bifurcation will necessarily be a scalar ODE A branch of solutions will be obtained for each 16 Setting so I Ak UL and linearizing at the point 11 uL yields EC span39r39k ODE Class Notes 68 The equations on WC must now be computed The graph of WC is given by u uL 77muL W07 s 622 Where W07 3 is the complementary direction ie Wms ZWOLQWWW 1 As a consequence of the center manifold theorem one has that aj0so Dsaj 0 so 0 Dnaj0so 0 Let lj be the eigenvectors of DfuT Which satisfy ltliu7 WW Sir Upon taking a Taylor expansion for at u uL and applying the operator lkuL one sees that ltlkuL7 u 4 UL ltlkuL7 DfuLu UL ltlkuL78u UL awn D2fltuLgtltu 7 up i As a consequence of equation 62 and the fact that 16011 W07 3 0 and since DfUL 8M UL 7780 7 87 kUL 2777 5 juL SM39WL 1 one sees that the ow on WC is given by l v 77 80 i 8 WWL D2fuLTkuL2gt772 0l3lllnl7 Z J 2 3 623 The claim is that z uLD2fuLrkuL vMuL To prove this rst note that lkltugtTDfltugtmltugt mu u e R Upon differentiating With respect to u evaluating at u uL and noting that DlEWL DfWLWkWL lguLDfuLDquot39kuL AMULXDlEWL kWL lEWLmWWL k ULDultlkuv 77k ugtl L 07 yields the desired result Equation 623 can now be Written as 1 v 77 So i S ltVAkuL7 TkuLgt772 0l5lll77lj7 2 1 2 3 Since the system is genuinely nonlinear it is not necessary to calculate the terms of 0lsli07lj for ij 2 3 Thus the bifurcation is of transcritical typel As an application of the lmplicit Function Theorem the critical points on WC are given by 77 0 u uL and l 8 80 ltVAMULgt7 TkuLgt77 0072 Let p0 gt 0 be suf ciently small and assume that ls 7 sol lt pol Upon parameterizing the above curve one has that for each lpl lt p0 there exists an 77R 77Rp and s 302 such that on WC 77 0 is connected to 77R at s s02 69 Todd Kapitula Now in order that the solution approach 77 0 as 2 A 700 one must necessarily have that s lt so By construction one has that UR UL 77Rquot39k39UIL W77R7 8 Upon performing a Taylor expansion for Ak at u uL and using the above expansion for UK one sees that AMUR MWL WAWLL WWLNUR 007 As a consequence one has that 8 AMUR WAMUL TMULMR 007 1 2 which since the system is genuinely nonlinear implies that s gt Ak UR The proof that the second Lax entropy condition follows from the strict hyperbolicity of the system will be left to the interested studenti D 62 Reduction to planar systems Now consider equation 61 in the case that dimEC 2 There are then three possible cases to consider a 06A0 0 and the eigenvalue is semisimple with multiplicity two b 06A0 0 and the eigenvalue has geometric multiplicity one and algebraic multiplicity two c 06A0 ii and each eigenvalue is simple In each of the cases enumerated above the ow on the center manifold will be described by a planar vector eld 621 The Hopf bifurcation Consider equation 61 under the condition that 0 0 is a critical point with 0 gZ 0D fmo As a consequence of the lmplicit Function Theorem for 1111 lt 11 there exists a unique curve of critical points 10 M with 10 moi Suppose that D fmn M has the simple eigenvalues am i i601 which satisfy a0 0 00 y 0 50 gt 0 624 Further suppose that 06D fzo0 ii 0i As discussed in 18 Chapter 202 the normal form for the equations on WC can then be written as i 1001 7 50 001 7 WWW f 0011571915 39 2 2 5 5 625 y BOW 00 b1 ayr y 01951 7191 A In polar coordinates equation 625 can be written as aw WM 005 6 26 9 MM WW2 004 Note that a T periodic solution to equation 625 is equivalent to having a solution Ttt9t to equa tion 626 which satis es 7 0 7 T 90 0 9T 27L Upon taking a Taylor expansion and neglecting the higherorder terms in equation 626 one nally gets the normal form equations to be studied r Dz0 a07quot3 9 50 OW MOW A careful study of equation 627 eg see the proof in 18 Theorem 2023 leads to the following result 627 ODE Class Notes 7 0 Theorem 66 Hopf Bifurcation Theorem Consider the system equation 61 under the constraints lead ing to equation 624 lfa0 y 0 and if lul is su ciently small then there eXists a unique periodic solution of0wll2gt Remark 67 The interested student should consult 18 equation 2012114 for an explicit expression for a0i 1f a000 lt 0 then the bifurcation to the periodic orbit is supercritical whereas if a000 gt 0 then the bifurcation is subcritical 622 The Takens Bogdanov bifurcation Now consider equation 61 under that assumption that f0u E 0 Furthermore assume that 0 C A0 has geometric multiplicity one and geometric multiplicity two As discussed in 18 Chap ter 206Chapter 331 the normal form associated with flow on WC is given by i y 2 628 y 1 M2yr 51 11 6 71471 Many interesting bifurcations occur in equation 628 however we will focus only on two Assume that 1 lt 0 When considering the critical point M1 0 the eigenvalues of the linearization are given by 1 Ai 7 i M 7 x7zfl i 2 7 W 7 Wm 7 6729 If one writes 2 xi l fie for 0 S 6 ltlt 1 ie m 7 7M 2mm 0amp2 630 then one can rewrite equation 629 as 1 Ai Elle i i 7u114 C62 Thus upon applying Theorem 66 one has that a Hopf bifurcation occurs at e 0 It can be computed that a0 1216 see 18 Chapter 2016 Since we are requiring that e gt 0 for the bifurcation to occur we must have that ill lt 0 hence it is supercritical If one assumes that b 71 then the bifurcating solution is stable whereas if b 1 the bifurcating solution is unstable Now let us rescale equation 628 in the following manner For 6 gt 0 introduce the scalings z I 62H y I 631 1 764 2 I 6212 t 2 es so that equation 628 becomes I dds u v 2 mm 1 71 u 612 0 buv When 6 0 equation 631 is a completely integrable Hamiltonian system with Hamiltonian 1 1 Hu v 2 E112 u 7 gugi The system has the homoclinic orbit uo tv0t where u0t 17 3sech2t i Via the use of Melnikov theory it can be shown that the homoclinic orbit persists for 12 212 06 49 2 m7g WW compare to equation 6130 71 Todd Kapitula l 1 1 1 u REFERENCES 1 W Boyce and R DiPrima Elementary Di erential Equations and Boundary Value Problems John Wiley amp Sons lnc7 6th edition 1997 A BroWder Mathematical Analysis An Introduction SpringerVerlag7 1996 J Carr Applications of Centre Manifold Theory7 Volume 35 of Applied Mathematical Sciences SpringerVerlag New York 1997 C Chicone Ordinary Di erential Equations with Applications7 Volume 34 of Texts in Applied Mathematics SpringerVerlag7 1999 5 WA Coppel Dichotomies in stability theory In Lecture Notes in Mathematics 629 SpringerVerlag 1978 6 J Hale Ordinary Di erential Equations Robert E Krieger Publishing Company lnc 2nd edition 1980 7 J Hale and H Kocak Dynamics and Bifurcations SpringerVerlag 1991 8 P Hartman Ordinary Di erential Equations John Wiley amp Sons7 lnc 1964 9 T Kapitula and P KeVrekidis BoseEinstein condensates in the presence of a magnetic trap and optical lattice Chaos 1530371147 2005 0 T Kapitula and P KeVrekidis BoseEinstein condensates in the presence of a magnetic trap and optical lattice twomode approximation Nonlinearity 1862491725127 2005 1 T Kapitula7 P Kevrekidis7 and B Sandstede Counting eigenvalues Via the Krein signature in in nite dimensional Hamiltonian systems Physica D7 1953amp42637282 2004 2 T Kapitula7 P Kevrekidis and B Sandstede Addendum Counting eigenvalues Via the Krein signature in in nitedimensional Hamiltonian systems Physica D7 2011amp2192017 2005 3 P KeVrekidis and D Frantzeskakis Pattern forming dynamical instabilities of BoseEinstein condensates Modern Physics Letters B 1821737202 2004 14 W Magnus and S Winkler Hill s Equation Volume 20 of Interscience Tracts in Pure and Applied Mathematics u u u 1 lnterscience Publishers lnc 1966 5 L Perko Di erential Equations and Dynamical Systems SpringerVerlag 2nd edition 1996 6 J Smoller Shock Waves and Reaction Di usion Equations SpringerVerlag7 New York7 1983 7 F Verhulst Nonlinear Di erential Equations and Dynamical Systems SpringerVerlag Berlin 2nd edition 1996 8 S Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Volume 2 of Texts in Applied Mathematics SpringerVerlag 2nd edition 2003 Math 571 Advanced Ordinary Differential Equations Prof Todd Kapitula Department of Mathematics and Statistics University of New Mexico December 3 2004 E mail kapitula mathunmedu 1 T Kapitula CONTENTS H Example BoseEinstein Condensates to Invariant Manifolds 1 Linear systems 22 The manifold theorems 23 Examples 231 Pitchfork bifurcation 232 Hyperbolic conservation laws 233 Bistable reactiondiffusion equation CA3 Melnikov s Method 31 Preliminary result 32 Homoclinic bifurcations 33 Example TakensBogdanov bifurcation g Hopf Bifurcation 41 Period doubling in maps 42 The Hopf bifurcation theorem 43 Example TakensBogdanov bifurcation o1 Maps 51 Linear maps 52 Invariant manifolds 53 Examples 531 Stable ow on WC 532 H non map Homoclinic points 541 The shift map 542 Transverse homoclinic orbits 543 544 Melnikov method revisited Example The forced damped Duf ng oscillator on Method of Averaging 61 Averaging 611 Example Duf ng equation 612 Example BoseEinstein condensates 62 Local bifurcations global behavior and Hamiltonian systems Comparison With a multiple time scales expansion 64 Almostperiodic vector elds 641 Preliminary estimates 642 The averaging theorem 643 Example BoseEinstein condensates Subharmonic orbits 651 Example KAM and Twist theorems 661 Algebraic preliminaries 662 KAM and Twist theorems 67 Example BoseEinstein condensates 533 w References Math 571 Class Notes 2 1 EXAMPLE BOSE EINSTEIN CONDENSATES A more complete description of the underlying physics as well as the mathematical formulation can be found in 18 19 21 Upon a suitable rescaling the quasione dimensional model for a BoseEinstein condensate with both a magnetic trap and optical lattice is given by 2x uh 47wq6ptcosjq76lql24 1 1 Hereq RgtltRgt gtC 397 1 2 1 2 21 L if 7281 21 V0cos M p R gt gt R is smooth and satis es pt T pt 6t E R and 6 E 7lll As discussed in 18 Section 2 one knows the following about a 0L Mjh o C R and each eigenvalue is simple with Mj lt 141 for all j 6 No b the eigenfunctions qj form a complete orthonormal basis and satisfy the decay condition lq z emzg S C I E R c ifj is even the eigenfunction is even whereas ifj is odd the eigenfunction is odd The desire is to construct a lowdimensional approximation to equation 11 Upon writing 4I7 75 N 50 U40 610341 I where cj R gt gt C and performing several rescalings one sees that the ODE governing the behavior of cj t is given by 8H 1539 7 J 65 where H is a particular Hamiltonian see 19 Section 2 for the details One feature of equation 12 is that 139 01 12gt lCotl2 l01tl2 327 173 which is a consequence of the fact that equation 11 conserves the total number of atoms Upon setting cj I 2pj em ie the pairs pj de ne actionangle variables one nds that 87 8H Pj7 j7aipja 10717 14 where now 1 l 7750707 p1 1507 4151775 0 WPO 1 Wm EPOXO EOPO Ola71 l l 6R2 iagoo g gain7i 043119 COS 2A gtPo 1gt Here A45 2 451 7 450 and R is a measure of the total number of atoms in the condensate Furthermore the coef cients are given by 21 11 I lt4i4jvqk4mgt7 1 I ltCOSIqznqjgtA Since equation 13 implies that p0 p1 E 1 after some algebra equation 14 can be reduced to p0 726R2p0l 7 p0 sin2Aq5 l M 7 7W1 7 0 7 elta 17 a50gtplttgt 15gt 6R2 aim 04300 ai11 0 13110 0052A 2P0 1 3 T Kapitula Note that equation 15 can be considered on the cylinder pmAqb 6 01 X 0270 Further note that if e 0 the system is autonomous hence7 it can be considered to be a periodic perturbation of a planar system Systems of such type are considered in detail in 9 and will be discussed herein Remark 11 In class I will present some numerical simulations of equation 15 with pt cost using the software Dynamics Solver The coef cients were derived upon setting it and V0 25 2 lNVARIANT MANIFOLDS For the system 139 10 10 the solution will often be represented as 45410 The function 4151 de nes a flow7 ie7 a sz s z 152 b 150 31 21 Linear systems First consider 139 AI 10 10 where A E Rn In this case it is known that 415 eA z Let 75 lt 7 E RUA be given7 and denote 05A A E 0A ReA lt 75 06A I A E 0A 75 lt ReA lt 7quot aquotA I A E 0A 7 lt ReA If 75 are chosen so that A E 06A if and only if ReA 07 then one has the usual de nition of these spectral subsets Associated with each spectral set there is a subspace ESgtCgtLl which satis es the property that eAtEscu C Escu Furthermore7 one can easily prove the following lemma Lemma 21 If 10 6 Es then there exists a constant M 2 1 such that leA zole VS S M7 t 2 0 If 10 E E then there exists a constant M 2 1 such that leA zole 7 S M7 t S 0 Finally if 10 6 EC then there exists a constant M 2 1 such that leA zolei l S M7 t 2 0 and leA zole VS S M7 t S 0 Remark 22 With the usual definition7 ie7 upon setting 75 7 6 with e gt 0 begin arbitrarily small7 one can paraphrase Lemma 21 to state that if 10 6 E57 then 415410 A 0 exponentially fast as t A 007 while if 10 E E 7 then 415410 A 0 exponentially fast as t A 700 If 10 E EC7 then the behavior has no apriori characterization Math 571 Class Notes 4 22 The manifold theorems Assume that for the ODE z39 one has that f R gt gt R is CT 7 2 2 with 0 De nition 23 A space X is a topological manifold of dimension k if each point z E X has a neighborhood homeomorphic to the unit ball in R Remark 24 In particular the graphs of smooth functions are manifolds De nition 25 Let N be a given small neighborhood of z 0 and let 6 gt 0 be suf ciently small The stable manifold W5 is W5 I z E N tze lst E NVt 2 0 and tze st A 0 exponentially fast as t A 00 The unstable manifold W is W I z E N tze E NVt S 0 and tzequot A 0 exponentially fast as t A 700 The center manifold WC is invariant relative to N ie if z 6 WC then tze l t 6 WC N t 2 0 and LIe lst 6 WC N t S 0 Furthermore WC N W5 WC N W Theorem 26 Stable manifold theorem There is a neighborhood N of z 0 and a Cpl function 5 N WES gt gt EC 69 E such that W5 graphhs Theorem 27 Unstable manifold theorem There is a neighborhood N ofz 0 and a CT 1 function N N E gt gt ES 69 EC such that W graphh Theorem 28 Center manifold theorem There is a neighborhood N of z 0 and a CT 1 function C N EC gt gt E GEES such that graphh is a WC Remark 29 One has that a dimWsgtCgt dimEsgtCgt b The manifolds are invariant ie if z 6 WW then E WsgtcgtLl for all t E R Thus if eg dimEs ks there is then a ksdimensional ODE which governs the behavior of the ow on W5 c WSgtCgtLl is tangent to ESgtCgtu at z 0 d The dynamical behavior on W5 and W is determined solely by the linear behavior e WC is not unique For example consider the system f The proofs of these theorems is discussed in 28 Chapter 335 The formulation of the manifold theorems has important implications First one can foliate W as well as W5 in the following manner Let A1k E aDf00 be simple and hence real be such that 0 lt A1 lt A2 lt lt Ak and let the associated eigenvector be denoted by vj By the statement of the unstable manifold theorem it is known that there is an invariant kdimensional manifold W which is tangent to Equot spanv1 vk at z 0 For a given 1 S E lt k let 7 be chosen so that Re Ag lt 7 lt Re A1L From the statement of the unstable manifold theorem there is an invariant manifold of dimension k 7 Z say WE which is tangent at z 0 to El 2 spanvg1 H4116 Note that W C W In this way one can foliate W with W135 C W132 CC Wf C W 5 T Kapitula Now consider the system i f177 A 07 where f00 0 and A E Rki Suppose that 75 and 7 are chosen so that aCltDflto0gtgt A e altDfltoogtgt ReA o and further suppose that dimEC Z As an application of the center manifold theorem there then exists an invariant kZdimensional invariant manifold so that the governing equations are of dimension kZi This reduced set of equations will determine all of the interesting bifurcations as it is known that any solutions on W5 W will have exponential behavior as t A 00 t A 700 Remark 210 Unless specifically told otherwise it will henceforth be assumed that 75 are chosen so that a Df0 A E aDf0 ReA 0 2 3 Examples The manifold theorems are also important in that they implicitly tell us how to compute the ow on W Consider the system y39 Ay91yz 239 B2g2yyz7 where y 6 ES 69 E 2 6 EC and igj y 0iy2i As a consequence of the center manifold theorem it is known that WC is given by the graph y hcz where RDhC Eel Since hcz is smooth it can be computed via a Taylor expansion The ow on WC is then given by 23 B292hczz as long as the trajectory stays in Ni Before considering the rst example we need the following proposition Proposition 211 Suppose that g z N C R gt gt R is C2 in a neighborhood ofz 0 and suppose that g0zg zn 0 for all 0zgi In C N There exists a neighborhood M C N and a 91 E C1M such that 91 11911 Proof By Taylorls theorem one has that 1 6 91 907127m71n a9ltt11712 7100 0 Choose M so that the line between I and 0zgi In lies in M This yields that E Eg zl 12 zn 11Dgtzl 12 zne1 which then implies that 1 911 Dgtzlzgiuzne1dt C1Mi D 0 Remark 212 In the next few sections we will go through a series of examples which will illuminate the power of the theory The interested student should consult 1 for a more complete discussion 231 Pitchfork bifurcation For the rst example consider the system i 71 12 7 y2 wwif 0 Q Math 571 Class Notes 6 The equation for e is appended to the system so that one can readily apply the manifold theorems At the critical point 00 0 one has that asDf0 1 Es swam a Df0 0 EC spane2esl Thus upon applying the Center Manifold Theorem one knows that the center manifold is locally given by z hye h0 0 hy00 h500 0 21 and the ow on WC is given by y39 6y yhy76 7 y3 0 The function hy 6 must now be determined It is clear that 006 is a critical point for any 6 E R hence h0e E 0 so by Proposition 211 one can write hye yhl yel As a consequence of the smoothness of the vector eld the function hl has a Taylor expansion which by equation 21 is given by haw ay be 0y2 62 Since WC is invariant one has that 7 I 7 Byy 86 67 which yields that 7hy e hy e2 7 y2 2ay be 9y2 62ey yhy e 7 ygl Simplifying the above expression gives a1y2 7 bey 0lyl 603 0lyl 603 which necessarily implies that In conclusion hy76 90y 0y2 62 As a consequence the ow on WC is given by y39 96 i 292 0lyl 603 2 2 6 0 Set My 6 I 6 i 292 0lyl 603 Since m00 0 and me 00 l by the lmplicit Function Theorem there exists an e 6y and yo gt 0 such that me yey E 0 for all lt yer By inspection one has that 6y 2y2 A depiction of the ow on WC as well as that for the full ow will be given in class The above example yields what is known as a pitchfork bifurcation 28 Chapter 20lle which has a normal form given by dc Iix2l 23 This normal form can be achieved from equation 22 by properly rescaling t and y and dropping the higherorder terms The removal of the higherorder terms is justi ed via the lmplicit Function Theoreml Another typical bifurcation associated with onedimensional center manifolds is the saddlenode bifurcation 28 Chapter 20llc which has a normal form given by iiz2l 24 The last typical bifurcation associated with onedimensional center manifolds is the transcritical bifurcation 28 Chapter 201d which has a normal form given by z39 zizl 2 5 7 T Kapitula 232 Hyperbolic conservation laws A viscous conservation law is given by u fux um 26 where f R gt gt R is Coo A thorough discussion of conservation laws and their importance in applications can be found in 277 Part Ill The goal here is to nd travelling waves7 which are solutions 112 2 I z 7 st of equation 26 which satisfy the asymptotics W a 27 In the travelling frame equation 26 is written as utisuzfuz um7 2 8 and the travelling wave will now be a steadystate solution7 ie7 a solution to 7 suz u um 29 It will be realized as a heteroclinic orbit The following hypothesis will be required Hypothesis 213 The function f satis es a for all u E R has distinct real eigenvalues A1ult mu lt lt Mu the system is strictly hyperbolic b ltVAjuTj lt 0 for all u E R where Tju is the eigenvector associated with the eigenvalue Aj u the system is genuinely nonlinear Remark 214 If n 17 the second condition is equivalent to specifying that is convex Lemma 215 For each uL E R and each 1 S k S n there exists a curve of states ul m for 0 lt p lt p0 such that a travelling wave exists With speed s 8k Furthermore a Mfg2 and 81607 are CT for any 7 E N and 139 k 139 k A pgg rd uLv 31907 MuL b AMul W lt SW7 lt AkWL C AkaWL lt 3160 lt Ak1u1 39 Remark 216 Conditions b and c are known as the Lax entropy inequalities Proof Set 39 I ddz lntegrating equation 29 from 700 to 2 and using the fact that A uL as 2 A 700 yie s 11 u fuL 3u uLl Linearizing at the critical point uL yields A I DfuL 7 311 Upon noting that altAgt A 7 s z A e altDfltuLgtgtL it is seen that a bifurcation can occur only if s Ak uL for some k 1 7n A branch of solutions will be obtained for each k Math 571 Class Notes 8 Upon appending s 0 to equation 29 and linearizing at the point uL so Where so I Ak uL one nds that EC spanTkuL0T 01Tl The equations on WC must now be computed The graph of WC is given by u uL mm Wow 210 Where W07 3 is the complementary direction ie WW 3 Z 1077 3TjuL jfk As a consequence of the center manifold theorem one has that aj0so Dsaj 0 so 0 Dnaj0so 0 Let 4W be the eigenvectors of DfuT Which satisfy ltZiu jugt Sir Upon taking a Taylor expansion for at u uL and applying Mk uL one sees that W 71L u 4 uLgt WWL DfuLu uLgt WWLL 8u uLgt l ZMuL ED2fuLu UL2gt As a consequence equation 2 10 and the fact that Mk uL W07 8 0 and since DfuL 8M uL 7780 7 5TkuL 2 777 8 A1 uL 8m 71L J k one sees that the ow on WC is given by 1 i v 77 50 i S WWW D2fuLTkuL2gt772 0l8l WL 1 J 2 3 s 0 The claim is that ZEuLD2fuLTk uL VAk To prove this rst note that k uTDfuTk Ak u u 6 Eur Upon differentiating With respect to u evaluating at u uL and noting that DZEuLDfuLTk uL E uLDfuLDTkuL AkWL MEWLW uL EWLWW 7 kuLDultzkuvTkugtluuL 07 yields the desired result The ow on WC can now be Written as 1 i v 77 80 877 ltVAkuL7TkuLgt772 0l8l WL 11 2 3 s 0 Since the system is genuinely nonlinear it is not necessary to calculate the terms of 0lslil77lj for ij 2 3 Thus the bifurcation is of transcritical typel As an application of the lmplicit Function Theorem the critical points on WC are given by 77 0 u uL and s so ltVkuL7TkuLgt77 0772 9 Ti Kapitula Let p0 gt 0 be sufficiently small and assume that ls E sol lt pol Upon parameterizing the above curve one has that for each lpl lt p0 there exists an 77R 77Rp and s 3p such that on WC 77 0 is connected to 77R at s 8pl Now in order that the solution approach 77 0 as 2 A Eco one must necessarily have that s lt so By construction one has that uR E uL 77RTkuL W77R7 8 Upon performing a Taylor expansion for Ak at u uL and using the above expansion for uR one sees that AkWR AMuL WMWL TkuLgt77R 007 As a consequence one has that l s 7 MuR ltVk 71L Tic uLMR 007 which since the system is genuinely nonlinear implies that s gt Ak 11R The proof that the second Lax entropy condition follows from the strict hyperbolicity of the system will be left to the interested student 1 233 Bistable reactiondiffusion equation Before we go to the example we first need a preliminary result For the system 139 10 Io recall that the ow is denoted by 45110 Let ys be a C1 curve such that y0 10 and let 1 I dyds0l By the chain rule tltyltsgtgtlso Dx zltzogtu so that Dw zo takes tangent vectors to curves of initial conditions to tangent vectors of the image Now upon using the smoothness of the ow one has t at d d Dx t10v Dx z0v Dmflt z10v Df z10Dx zIov hence D tzov solves the equations of variation 539 Dflt zlt ogtgt The utility of this result is that one now has a way of following the ow of vectors tangent to the stable and unstable manifoldsl Now the techniques and ideas covered in this example are not presented in 28 The interested student should consult 12717 20 and the references therein for a small subset of other examples Consider u um u 211 where 1 fu uuia17u 0lt a lt if It can be shown that u E 0 and u E l are attractors for equation 211 whereas u E a is unstable In the context of mathematical biology in which equation 211 is known as Nagumols equation the solution u E 0 represents the rest state of a neuron the solution u E 1 represents the active state of a neuron and uz t is the voltage across the membrane The goal will be to find travelling waves 112 2 I z E ct which are transitions between the two attractorsl The wave can be thought of as approximating the front of the nerve impulse The interested student should consult 376 11 and the references therein or the web site of James Murray for more details on the biological applicationl Math 571 Class Notes 10 1n travelling coordinates the wave must satisfy iieufu0 39 7 2112 Which as a system can be Written as u v 2113 139 70v 7 Note that if a wave exists then the wave speed must satisfy 57 mug 701 fudu Which since 0 lt a lt 12 implies that e lt 0 Lemma 217 There exists a unique 0 lt 0 for Which equation 213 has a solution Which satis es A 0 2 A 700 1 2 A 001 Furthermore the solution is unique up to spatial translation Proof The critical points of equation 2113 are 0 0 a 0 and 1 0 The desired travelling wave satis es 00 2 A 700 10 2 A 001 1437142 A Linearizing at 0 0 gives the eigenvalues A z gm W and linearizing at 1 0 gives the eigenvalues 1 Ali Epci J52 7 4M1 Since f 0f 1 lt 0 one has that both Atil lt 0 lt A0111 Thus for a xed value of 5 there exists a onedimensional W0 1 and We at 00 and 10 and these manifolds are tangent to Eal and Eal Where 1931 Span lGJQDTlv 1931 Spanf Jof Tl Note that the desired travelling wave must satisfy W3 0 f g for some 0 lt 0 Set Luv uav20 Kuv 0 u av20l One has that 1 A01T points into K thus as a consequence of the unstable manifold theorem W0 0 enters n examination of the vector eld on 8K yields that W C can leave C only by crossing L or by having 1 A 001 Since a v the second scenario is precluded Furthermore since u v on L one can have that W0 0 intersects L only once Finally W0 5 N L f g for otherwise an application of the Poincare Bendixson theorem yields a closed orbit in K Which is impossible because int K contains no critical points Similarly intersects L and does so only one time As a consequence of the fact that Ag and A are smooth in c it can be shown that W0 5 and are smooth Set g e I W0 e N L 950 2 Ll 11 T Kapitula These functions are wellde ned and smooth furthermore a travelling wave exists whenever g c 950 First suppose that c 0 The system then has a rst integral given by Euv 2 112 A dsi Since a lt 12 one sees that g 0 lt 950 Now x a E RJF and set Kaz uv 0 u 1 vzau There exists an M E R such that lt M for u 6 01 thus on the line 1 au one has that 39 M 3 i i M gt 707 ii u au 1 5 Choose 51 lt 0 so that if c lt 51 then gt a For these values of 5 one then has that the vector eld points into K04 on the line 1 aui Now choose 52 so that AOC gt a for c lt 52 If C lt minElEg then W0 5 C K04 and hence intersects L inside of Cal Now Wf 5 cannot enter K04 for otherwise the trajectory on could not go to 10 as 2 A 001 Thus 955 lt g c for c lt minElEgi By the Intermediate Value Theorem there then exists a 0 lt 0 such that gsc g ci It must now be shown that 0 is unique First it is an exercise to show that g c gt 950 for c E Rf The uniqueness for c E R will follow if it can be shown that g c and 950 are monotonei Append c39 0 to equation 2 13 let 5 I 6u 6v 60T and recall that the equations of variation 6216v 657fu6u706v7v60 550 carry tangent vectors to tangent vectorsi One solution to the variational equations is 51 I it i 0T where 112 vzcT is any solution to the ODE Let 52 be any other solution to the variational equations such that 51 52 is a linearly independent set We will later set 52 to be tangent to either Wowz U W0 c or W552 U Wfci CER CER The vector 51 X 52 is perpendicular to span 1 2 hence we will compute an ODE for this vector In particular we will compute the ODE for w I 51 X 52 egi Since 56 0 without loss of generality one can set 52 yl y21Ti This then implies that w uyg 7 byl which upon using the variational equations gives 71 7192 1M1 W2 i191 75w 7 v2 recall that it 1 Suppose that dg 0 0 52lt gt lt 7 dc iiei 520 is tangent to g c and hence Wocui Since 51 and 52 are tangent to W06 there exists a C E R0 such that 71T7 lim 51 X mew CAg 710T in particular 7 l1m we 0 z 0 zaioo Since vze 32 01 as 2 A 700 this yields that ieicz ecsv2s dsi Math 571 Class Notes 12 Since d wlt0gtvltogt one nally sees that d u 1 0 C im ecsv2s ds lt 0 In a similar manner one nds that dgs 1 00 05 2 7 e v 8 ds gt 0 as w lt Thus both curves are monotone so that 0 lt 0 is unique D Remark 218 Note that when 0 0 g g5 with d u s 1 T as 2 i 7 if d dclt9 9 van7 e v s 3 When discussing Melnikov theory it Will be important to understand and evaluate quantities such as that given above Corollary 219 Consider ii0ufu6gu1i6 0 Where 9 R X R X R gt gt R4r is smooth There exists an 60 gt 0 such that ifO S 6 lt 60 then there exists a unique travelling wave With speed 06 Where 06 is smooth With 00 0 Proof Construct the curves 9 06 and 9506 as in the proof of Lemma 217 It has already been seen that d yquotc70 95370 07 CTlt9 c70 95C70 lt 0 0 By the lmplicit Function Theorem there exists an 60 gt 0 such that for 0 lt 6 lt 60 there is a smooth 06 With 00 0 such that yquot067 6 7 95C67 6 0 D 3 MELNIKOV7S METHOD In this section we Will consider another technique to show the r 39 t of t quot quot 39 orbitsl As in the example of Section 233 the result Will depend heaVily upon the lmplicit Function Theoreml The material presented in this section is also covered in 28 Chapter 28 31 Preliminary result Consider the system 9 Atyyt7 31 Where a At E R2 gt E R2 are continuous and gt is uniformly bounded b lim At A0 With the approach being exponentially fast up c trace At E 0 not an essential assumption d A0 Pdiag 7P 1 Where A E Rl 13 T Kapitula Now assume that there exists a solution yht to the homogeneous system 9 7 Amy 32gt such that lyhtle S C iiei yh t decays exponentially fast as ltl A 00 De ne dv1 i vn I detv1i i vn v1 i vn E R and let yg t be the solution to equation 32 which satis es dyh 0 yg 1 As a consequence of Abel s theorem one has that dyh t yg E l which implies that ly2tle 2 C iiei y2t grows exponentially fast as ltl A 00 We shall now construct a special set of solutions yi t to equation 31 In particular we wish to have y t y be bounded for t 2 0 t S 0 Upon applying variation of parameters to equation 31 one can check that the desired solutions are given by yilttgt7ciyhlttgtyhlttgt dltgltsgty2ltsgtgtdsy2lttgti dltyhltsgtgltsgtgtds 33gt where Ci E R It can be checked that the assumption on yh t guarantees that these solutions are appropri ately bounded furthermore if lgtl A 0 suf ciently fast as ltl A 00 then lyitl A 0 as t A iooi As an observation note that 00 y 7 mm 7 a 7 mm 7 mo dltyhltsgt79ltsgtgtdsi Now consider the adjoint equation to equation 32 239 7ATtzi 34 Let 2h t and 22 t be two solutions which satisfy lt2207yh0gt17 lt2207y20gt 0 ltZh07yh0gt 07 ltZh07920gt 1 Since lt2tytgt C for any solutions yt to equation 32 and 2t to equation 34 one has that ZQWHM 2 C zhwew g a Furthermore one can rewrite the result of equation 33 to say that 00 Wu J 7 mm 7 dltyhltsgt79ltsgtgt as 35gt 32 Homocinic bifurcations Consider the system i f1691767 where f R2 gt gt R2 and g R2 X R gt gt R2 are smooth Suppose that a 0 and z 0 is a saddle point b V fz E 0 c when 6 0 there exists an orbit q0t homoclinic to z 0 The goal of this subsection is to consider the persistence of go t under the perturbation Before doing so we need the following preliminary resulti Math 571 Class Notes 14 Lemma 31 There is an 60 gt 0 such that for all 6 lt 60 there eXists an 15 With 10 0 such that f1e 691576 0 Furthermore for each 6 lt 60 there eXists onedimensional manifolds W 15 and WS 6 and these manifolds are smooth in 6 Proof The persistence of the critical point follows from setting we M ems noting that h0 0 0 and D h00 Df0 is invertible and invoking the Implicit Function Theorem The existence and smoothness of the manifolds follows from the manifold theorems 1 In order for the homoclinic orbit to persist one needs that W 15 Wse g We now proceed to make the calculation that will ensure this intersection Let represent the trajectory on W 15 Ws 6 and recall that by assumption 13 t EEO qo As a consequence of smoothness one can write 2amp0 aw 61W 062 t 0 16 aw 602W 062 t2 0 where the functions 155 are uniformly bounded on their domains Upon noting that z a z2gtltogt so 7 mm 08 36 one sees that a necessary condition for persistence is that 7 0 Since the manifolds are invariant one has that fzi waits thus upon differentiating with respect to e and evaluating at e 0 one sees that i Dfqo 9 0707 t 0 ii Dfqo 9 0707 t2 0 These equations must be solved so that 151 quot are uniformly bounded on their respective domains Since one solution to the homogeneous equation y39 Dfqoy is yh do fqo As a consequence of the discussion in Section 31 one has that Ii 9590 6400 M9203 37 where the Melnikov function M is given by 00 M df40579 1087 0 as 38 700 Now as in Section 31 let 2 t be the solution to the adjoint equation 2 7DfqoTz which satis es ltZh07400gt 0 in particular note that 2 0 is orthogonal to the vector eld As a consequence if M 140 Ii IMO 39 is such that M 0 then up to C62 one has that W ze Wsze g Theorem 32 Let M R gt gt R be de ned as in equation 38 and suppose that Mu0 0 With Muo 0 There then eXists an 60 gt 0 such that for all 6 lt 60 there eXists a unique ue With u0 uo such that W 15 Wsze g for u ue 15 T Kapitula Proof As a consequence of equation 36 and equation 39 we have that 140 I I0gt MM 06 The result now follows upon applying the lmplicit Function Theorem to 90 e 2 MW 06 D Remark 33 One has that a if V f 0 then the Melnikov function is given by 00 S M doV w drdltfltqoltsgtgt79ltqoltsxwas b the theory is also applicable when studying the persistence of heteroclinic orbits c there is an analogous theory for the system i f1 6917t757 where gzt Te gzt e for some T gt 0 see 28 Chapter 28 and 9 Chapter The last remark is especially pertinent and the implications of periodicity in the forcing function will be explored more fully at a later date 33 Example Takens Bogdanov bifurcation Consider i fIM and assume that f0t E 0 Furthermore assume that for A I D f00 one has that a A 0 and that this eigenvalue has geometric multiplicity one and algebraic multiplicity two As discussed in 28 Chapter 20 6Chapter 331 the normal form associated with ow on WC is given by z39y 39 2 310 y 1 M2yr 51 11 E 711i For 6 gt 0 introduce the scalings 1 e2u y I 631 1 764 2 I 6212 t 2 65 so that equation 3 10 becomes it v 311 i 71 u2 612 0 buv When 6 0 equation 311 is a completely integrable Hamiltonian system with Hamiltonian 1 l Hu v 2 E112 u 7 gugi The system has two critical points the point 10 is a saddle point which has the homoclinic orbit uo t v0 t where u0t 17 3sech2t and the point 710 is a nonlinear center The result of Theorem 32 can now be used to determine the persistence of the homoclinic orbit The Melnikov function is given by MW W m vac at b M uolttgtv3lttgt at 712 7 512 Math 571 Class Notes 16 Hence the orbit will persist for 12 Vg where 5 121 2 05 Note that for 12 lt 12quot M lt 0 while for 12 gt 15 M gt 0 This yields the relative orientation of the stable and unstable manifolds Further note that with respect to the original variables the orbit persists for 49 2 M1 g 0W 7 and that zt OOLQ 4 HOPF BIFURCATION In Section 33 we considered one type of planar bifurcation in particular the existence and persistence of homoclinic orbitsl Herein we will now consider the creation of periodic orbits via what is known as a Hopf bifurcation The technique behind the proof will require knowledge of a particular type of bifurcation associated with maps as we will construct the bifurcating periodic solution via a Poincare map 41 Period doubling in maps Consider the discrete dynamical system zn1 rms 41 where f R X R gt gt R is a smooth diffeomorphisml Using the notation fk f o f o f k times we say that 7izo I 10 fi1zo e l l l fikzoel l are the positive and negative orbits of 10 and 710 2 7 zo U 7zo is the orbit of 10 We say that y E wzo if there is a sequence with lim ni 00 such that y lim f 1zoe and the set wzo is the union of all such points The set azo is de ned by letting limni fool Finally a set M C R is invariant if fMe M ie if for each I E M one has that fze E M and if for each y E M there exists an I E M such that fz e yr One important example of an invariant set is a periodN orbit 10 fzoe l fNzoe in which 10 fN1zoe but 10 fjzoe for any 1 S j S Ni In order to see how such an orbit can arise consider the linear map a b In1AIm A1 lt 7b a gt7 where a2 b2 1 If I 11 12T then upon writing 11 I Tcost 12 I Tsint one sees that Tn1 Tm 9n1 9n 40 where w 6 0270 is de ned so that a ib emf Thus the circle 7 7 0 is invariant and the mapping on this circle is 9 90 7 mar lf 27rw E Q then for some N E N one has that Nw 0 mod 27f which implies the existence of an Nperiodic orbitl lf 27rw E RQ then the orbit on the circle is dense ilel given a point 9 there is a sequence with limnj 00 such that limt90 7 njw 9 mod 2 Now consider equation 41 in the scalar case and assume that f0 e E 0 with f7 0 0 ill A Taylor expansion then yields 1 l fze7z a12bezgczg If one tries to nd nonzero xed points of 06 then one is left to solving 1 1 17z a12bezgczg 17 T Kapitula which clearly has no solutions of 0lel12 If one now tries to nd xed points of l l f216172b617 ca213n7 then one is left to solving 1 1 z 172b617 ca213 i If I f 0 then as an application of the lmplicit Function Theorem one has that nontrivial solutions lie on the curve 03 a2 412 7 where S is known as the Schwarzian derivative Thus xed points for f2z e arise via a pitchfork bifurcation If S gt 0 the bifurcation is subcritical whereas if S gt 0 the bifurcation is supercriticali Let the upper curve be denoted by z z e and let the lower curve be denoted by z If One has that e 7512 C13 S I 6 f2r 6767 126 f2ri6767 which in turn implies that f1quot676 f31quot6767 fI 76 0520976 Since f3 f2 o f this yields that fz ee and fzee are also xed points for f2zei Since fze has no xed points other than I 0 by uniqueness one must have that 126 f1quot6767 6 fI 76A Thus under the conditions f 00 71 and b fez 00 0 we have shown that a period2 orbit given by 1quot e fz e 6 arises via a pitchfork bifurcationi Remark 41 An alternative interpretation of the above result is that xed points for f2 arise via a pitchfork bifurcationi These points arise i 62 8186 f206 E 0 f200 1i 42 The Hopf bifurcation theorem Consider z39 fIM where f R X R gt gt R is smooth Let 100 be a critical point such that 0 0D fzo0i As a consequence of the lmplicit Function Theorem for ml lt M there exists a unique curve of critical points z with 10 10 Suppose that D fzt has the simple eigenvalues a i i which satisfy 10 0 00 a 0 50 gt 0 42 Further suppose that aCD fzo0 ii 0i As discussed in 28 Chapter 202 the normal form for the equations on WC can then be written as i 1001 7 50 001 7 WWW f 0lrl57 M5 2 2 5 5 43 y BOW 00 0001 WWI y 0lIl vlyl A In polar coordinates equation 43 can be written as aw WW 005 4 4 9 MM WW2 004 Note that a Tperiodic solution to equation 43 is equivalent to having a solution Tt 9t to equation 44 which satis es T0 7 T 60 0 6T 27L Math 571 Class Notes 18 Upon taking a Taylor expansion and neglecting the higherorder terms one nally gets the equations to be studied T Dz0 a0T3 9 50 OW MOW Theorem 42 Hopf Bifurcation Theorem Consider the system equation 43 under the constraint given in equation 42 lfa0 0 and if M is su ciently small then there exists a unique periodic solution of OWN2 45 Proof For the initial condition T0 E 90 0 denote the solution to equation 45 by 7 t5 and 90254 Since 20 E 07 927r 70w E 2m upon applying the lmplicit Function Theorem one obtains a neighborhood U0 gtlt V0 of T M 00 and a smooth function T U0 gtlt V0 gt gt R With T00 27r 0 such that 9TEMEM E 27L De ne the Poincare map H U0 gtlt V0 gt gt R by 115401 TT57M757M7 and recall that xed points of H yield periodic orbits for equation 43 The goal is to show that H has the same properties as the map f2 described in Section 41 in particular that it undergoes a pitchfork bifurcation at the point T M 00 First note that P0M E 0 and that 8H dT dT 750370 T5070T 35070 Since T 0 is invariant one has that T 0 Furthermore When T 0 one gets that dT dT dT 01 1 OWCTE 50 17 Which for M 0 has the solution BTBEQ 1 Hence 6H 970 0 7 1 A tedious calculation reveals that When T M 0 0 33amp 8M 85 7 8M 85 When T 0 one has that g in We dt at as a as Which implies that E 8H 1 0 5M 65 60 0 The Poincare map then undergoes the desired pitchfork bifurcation Which implies the existence of a xed point of Owl2 The uniqueness of the xed point follows from the fact that a0 0 see 28 Chapter 202 The interested student should consult 28 equation 20214 for an explicit expression for a0l 1 Remark 43 To leading order the Poincare map is given by 10 60 lf a0o0 lt 0 then the bifurcation is supercritical Whereas if a0o0 gt 0 then the bifurcation is subcritical Remark 44 An alternate proof of Theorem 42 is given in 28 Theorem 2023 new 1 m gt5 alt0gt53 19 T Kapitula 43 Example Takens Bogdanov bifurcation Recall the system given in equation 310 Assuming that 1 lt 0 when consider the critical point ix7M1 0 the eigenvalues of the linearization are given by At MixTMilMimyiswle If one writes M2 47 56 for 0 S 6 ltlt 1 ie M1 6 2M2M 0627 then one can rewrite the above as 1 Ai E 36 i i C62 hus upon applying Theorem 42 one has that a Hopf bifurcation occurs at e 0 It can be computed that a0 1216 see 28 Chapter 206 Since we are requiring that e gt 0 for the bifurcation to occur we must have that bit lt 0 hence it is supercritical If one assumes that b 71 then the bifurcating solution is stable whereas if b 1 the bifurcating solution is unstable 5 MAPS While the study of general maps is interesting in its own right we will primarily focus upon the properties associated with Poincare maps The eventual goal of this section is to rigorously study the dynamics with periodically forced Hamiltonian systems One concrete example of such a system is the periodically forced and damped pendulum which was discussed in some detail in Math 512 51 Linear maps First consider the scalar linear map In1 aIn a E C which has the solution In anIo Upon writing a I lale for some w 6 0270 one sees that a neinwzo In l Thus it is seen that if lal lt l the solution has the behavior that In A 0 as n A 00 while if lal gt 1 the solution has the behavior In A 0 as n A 700 lal 1 then as was seen in Section 41 the solution will either be periodic or its trajectory will densely ll the circle with radius I0l Now consider In1 AIn A E Rnxn 51 which has the solution In AnIo Analogously as in Section 21 but with less generality set 05A I A E 0A Al lt1 06A A E 0A Al 1 a A I A E 0A Al gt1 Associated with each spectral set there is a subspace ESgtCgtLl which satis es the property that AEscu C Escu Let 0 lt 6 ltlt 1 One can easily show that for equation 51 that there is a CMe 2 l and a 6 01 such that if Math 571 Class Notes 20 a 10 6 Es then lznla n S Clzol n 6 No b 10 E E then lznlan S Clzol in 6 N0 c 10 6 EC then Me 117 e lzol S S Mel e lzol n E Z 52 Invariant manifolds Let us now consider the mapping zn1 52 Where f E ORR R 7 2 2 is a smooth diffeomorphism Note that the solution is given by In f 10 It Will be assumed that 0 so that z 0 is a fixed point of the map De nition 51 One says that A 0 exponentially fast if there exists an a 6 01 and a C E R such that lf zla n S C for n 6 N0 De nition 52 Let N be a given small neighborhood of z 0 and let 6 gt 0 be suf ciently small The stable manifold W5 is W5 I z E N z E NVn 6 N0 and A 0 exponentially fast as n A 00 The unstable manifold W is Wu I z E N z E NVn 6 N0 and A 0 exponentially fast as t A 700 The center manifold WC is invariant relative to N ie if z 6 WC then E N for n 6 No Further more WC N W5 WC N W 0 and dim WC dim W5 dim W n Theorem 53 Stable manifold theorem There is a neighborhood N of z 0 and a Cpl function h5 z N WES gt gt EC 69 E such that W5 graphhs Theorem 54 Unstable manifold theorem There is a neighborhood N ofz 0 and a CT 1 function h z N N E gt gt ES 69 EC such that W graphh Theorem 55 Center manifold theorem There is a neighborhood N of z 0 and a CT 1 function hC N N EC gt gt E 69 Es such that graphh is a WC Remark 56 One has that a dimW5gtCgt dimEsgtCgt b The manifolds are invariant ie if z E Wsgtcgt then E WsgtcgtLl for all n E Z Thus if eg dimEs ks there is then a ksdimensional map Which governs the behavior of the ow on W5 c Wm is tangent to Em at z 0 d The dynamical behavior on W5 and W is determined solely by the linear behavior e WC is not unique f The proofs of these theorems is discussed in 28 Chapter 335 5 3 Examples In order to fully illustrate the utility of the manifold theorems consider the following examples 21 T Kapitula 531 Stable flow on WC Letting z I y 2T consider fyz lt A6 01 For the xed point 0 0T one has that as A Es Span170T ac 1 EC Span071TA There is a onedimensional Ws and a onedimensional W What are the dynamics on WC The manifold WC is given by the graph y hz Where hz a22 C23 Since WC is invariant upon using the equations one sees that hl Ahz 7 22 Which eventually implies that 1 hz71 22 C23 Thus on dynamics on WC are given by l A 7 1 Since A 6 01 one has that lt for f 0 suf ciently small hence the xed point 2 0 is W c 2n1 Czn Cz 2 21 22 C23 asymptotically stable on 532 H non map Letting z I w 2T consider 397 l 2 7 A102 fwz 17 lt 3w gt The xed points are given by 10 2 Where 1ix1A w I A 2f 2 3w When A 3 the xed point w2T 713 7lT is such that aquot 3 E span171T ac 1 EC span1r3TA Thus at this xed point there is an associated quot 39 W and quot 39 WC What are the dynamics on W As a consequence of the discussion in Section 41 one expects a pitchfork bifurcation to a period2 orbit on W In order to determine the stability of this orbit one needs the equations Which describe the dynamics on W So that the behavior on WC can be more easily analyzed rst put the system into canonical form via 397 397 l 1 z 397 1 wl3 p473 1 map lt y Under this transformation the system becomes 1 1 1 2 In1 lbw 22 i you gwn 7 3 Wain yn2l l l l yn1 ll g dr Zl n 62 glow 33 MIn Math 571 Class Notes 22 The manifold WC is given by the graph y hz M where h0 0 has 0 0 0 Upon using the invariance of WC it can eventually be shown that the dynamics on WC are given by 1 4 3 27 In1 905me 90541 1 EM 1 EM 112 E13 C14 Since 25 9 2 7 7 7 7 7 3 4 G 967041 88095 41 00 7 upon following the analysis in Section 41 it is seen that on WC there is a supercritical pitchfork bifurcation to an unstable period2 orbitl 54 Homocinic points Much of the material discussed in this section is also discussed in 28 Chapters 23 24 We will rst discuss the dynamics associated with a relatively simple mapl After this task has been accomplished we will relate these dynamics to those associated with a seemingly more complicated probleml 541 The shift map Let A 2 01 and let 5 denote the collection of all biin nite sequences of elements of A ie ifs E S then s quot39Sn39quot8723711808132quot39Snquot39 sj E A For s E 5 de ne the distance between 8 and via 00 A A 6 0 3n 5n d s s z 7 6 I A gt 2 W 1 8n y and note that ds S 3 One then has that ds is small if the two elements agree on a suf ciently long central block 28 Lemma 2412 De ne the homeomorphism a S gt gt S by 03 2 sn1 ie 03 S28718018182 The mapping a is known as the shift mapl Biin nite sequences which repeat periodically will be denoted by the nite length sequence with an overbar ie 010101010101 It is easy to see that am m mm hence a has two xed points In addition one can check that mm m Since cam m we have the existence of a period2 orbitl It is clearly not dif cult to construct periodn orbits for any n E N It will next be shown that a has an uncountable number of nonperiodic orbitsl Consider the bijective mapping between biin nite sequences and in nite sequences given by 54 3234303132 sn gt gt 50515432343334 snsn Every 7 6 01 can be expressed in base 2 as a binary expansion via 00 6n HZ swam 711 23 T Kapitula lf 7 E 01Q the sequence 1516263 is nonrepeating Since 0 is uncountable we now have the existence of an uncountable number of elements 8 E S such that 03 is not a periodic orbit It will nally be shown that there is an element 8 E 5 whose orbit is dense First construct all possible sequences having length n E N eg length 1 z 0 1 length 2 00 01 10 11 For a given length n there will exist 2 possible sequences Order these sequences in the following manner Let S81Sk 1 j We say that s lt if k lt j and if k j then s lt if 81 lt where Z 6 N is the rst such integer that 81 Denote the sequences having length k via 8 lt 31126 lt lt 812 Now set 7 33332211223333 8d 7 3335343284328251318331833587 and note that 8d contains all possible sequences of any xed length Let s E S be given let 6 gt 0 be given and let N E N be such that Writing Squot3957Nquot39371 30quot39SNquot397 we have by construction that there exists an M E Z such that O39M Sdj Sj for S N which implies that daM 3ds lt 6 Since 8 E S is arbitrary as is e gt 0 the orbit of 8d is dense in 5 Lemma 57 The shift map a S gt gt S has a a countable in nity ofperiodic orbits of arbitrarily high period b an uncountable in nity of nonperiodic orbits c a dense orbit Remark 58 One has that a While we constructed one point which has a dense orbit upon rearranging the ordering of the blocks 8 in 8d it can be clearly seen that there are at least a countable number of points which have a dense orbit b It can be shown that S is a closed perfect ie every point is a limit point and totally disconnected set 28 Proposition 2414 In other words 5 is a Cantor set c In the above we considered the shift map on two symbols There is a discussion of symbolic dynamics on N symbols in 28 Chapter 24 Let s E S be given and let 6 gt 0 be given Let 6 S be such that ds lt e furthermore suppose that Sj j for j 7N N Now suppose that 3N1 N1 It is clear that daN1s UN1 2 1 hence for the given point 8 and for each small neighborhood of 8 there exists an uncountable number of points such that after a xed number of iterations these two points are separated by a xed distance A system displaying such behavior is said to exhibit sensitive dependence on initial conditions De nition 59 A dynamical system which displays sensitive dependence on initial conditions on a closed invariant set which consists of more than one orbit will be called chaotic Math 571 Class Notes 24 542 Transverse homoclinic orbits De nition 510 Let f R2 gt gt R2 be a smooth diffeomorphism with a hyperbolic xed point p lf Wsp N W p f g then we say that p is a transverse homoclinic point if for some 4 E Wsp N W p Tqup EB TqWquotp R2 Remark 511 One has that a For ODEs a homoclinic point cannot be transversal as z39h E Tqup TqW p where zht is the homoclinic orbit b If p is part of an orbit of period 16 then one can discuss transverse homoclinic orbits for p simply y considering the mapping Theorem 512 Let f R2 gt gt R2 be a smooth diffeomorphism With a transverse homoclinic point p For any neighborhood U ofp there exists a Cantor set A C U and an n E N such that the mapping f z A gt gt A is topologically conjugate to the shift map on N symbols for some N 2 2 Proof See 28 Chapter 26 and 9 Chapter 5 1 Remark 513 One has that a mce 1s opo ogica y equiva en 0 e s 1 map in a emp mg 0 un ers an e ynamics or S39 n39 t l 39 ll 39 l tt th h39ft 39 tt t39 t d t dth d 39 f f it is suf cient to study those of the shift map b The set A is a hyperbolic invariant set Now suppose that f R2 gt gt R2 is a smooth diffeomorphism and that 100101 pn p0 are hyperbolic xed points Further suppose that W pj Wspj1 transversely for j 0 n 7 l The xed points along with their stable and unstable manifolds are then said to form a heteroclinic cycle One can show that under this scenario W pj Wspj transversely for each j 0 n 7 1 Thus as a consequence of Theorem 512 one has that there are invariant Cantor sets Aj in a neighborhood of pj for j 0 n 7 l on which the dynamics are equivalent to those of the shift map 543 Melnikov method revisited We must now determine a concrete set of examples to which the result of Theorem 512 can be applied Consider i M em 53gt where f R2 gt gt R2 and g R2 X R gt7 R2 are smooth and gzt T gzt for some T gt 0 Assume that when 6 0 a there exists hyperbolic critical point p0 which is a saddle point b there exists an orbit qo t homoclinic to p0 c EU The last condition is automatically satis ed if the unperturbed system is Hamiltonian For each to E 0T set 210zt t to and note that there exists a Poincare map H10 EEO gt gt 210 which is de ned by the timeT map When 6 0 one has that H10 p0 p0 furthermore since p0 is a hyperbolic saddle point for each 6 gt 0 there exists a pa p5 to with lim n0 p5 p0 such that H10 p5 p5 Finally for each to the point p5 is a hyperbolic saddle point for H10 so that there exists a onedimensional W p5 and Wspe Note that by supposition when 6 0 W p0 Wsp0 g however the intersection is not transverse 25 T Kapitula We Wish to determine a condition Which for e gt 0 Will not only guarantee that W pe Wsp5 g but that the intersection is transverse Let 75t be the T periodic orbit Which satis es 75 to per There exists stable and unstable manifolds for 75 t Which are given by Wquot 1 U WquotPe7 WSW 1 U WSltPel LOER to ER Suppose that 40t intersects the line 12 0 transverselyi Now W 75 N 12 0 is given by the curve 11 h t e and Ws 75 N 12 0 is given by the curve 11 hstei Upon setting 0w 2 We 7 W e one has that the desired transversal intersection Will occur if and only if for some to E 0 T Ct0e 0 Gt0e a 0 54 Since Ct 0 E 0 one has that Ct e E t e The Melnikov function is given by Mt t0i As an application of the lmplicit Function Theorem one has that if MW 07 M Oo 07 55 then equation 54 is satis ed hence if equation 55 is satis ed the relevant manifolds intersect trans versely and the result of Theorem 512 applies to Poincare map associated With equation 53 In a manner similar to that Which led to Theorem 32 it can be shown that the Melnikov function is given by Mae 2 0 dltfltqolttgtgtgltqolttgtt to at 56 also see 28 Chapter 28 and 9 Chapter 45 Remark 514 The interested student should compare equation 56 With equation 38 544 Example The forced damped Duffing oscillator The forced damped Duf ng oscillator is given by i y 7 3 57 y 7 z 7 z 67coswt 7 6y When 6 0 there exist homoclinic orbits L10i t Which are given by i i 7 10 t 397 i secht 40 t 7 lt yoi quot6 Fsechttanht Upon using equation 56 one sees that the Melnikov function for each orbit is given by 00 00 Mit0 76 y0it2 dtify y0it coswtt0 dt 4lt6i Rltgt39 t M 3 hf 77 wsmw w 277rwsec if 3 7 0 Ni 2 A plot of critical surface 6 Rw7 is given in 28 Figure 28i5i2i One has that if 67 lt Rw then there exists a to E 027rw such that Mt0 0 With Mto 0 see 28 Figure 2851 Hence in this case there are chaotic dynamics associated With equation 57 Conversely if 67 gt Rw then Mt0 0 for all to E 0 27rw so that the manifolds do not intersect Hence in this case one cannot conclude the existence of chaotic dynamicsi Math 571 Class Notes 26 6 METHOD OF AVERAGING Most of the material considered in this section can be found in 9 Chapter 4 also see 26 Consider the weakly forced nonlinear oscillator given by iw z efzit 61 Where fzz39tT fzi7 t R X R X R gt7 R is smooth7 and 0 S 6 ltlt 1 Suppose that T 27rw for some w E R i For a given k E N set 397 cosqb 7kwsinq 397 wit A 7 lt 7sinq5 7kwcosq5 gt qb i ltZgtAtltgtl 62 Note that At is periodic With period Zakml Under this transformation equation 61 becomes a 5 Ieflt17i7t sinqb LA 139 5 Ieflt17i7t cosqbl LA and de ne 63 In equation 63 one has that z zu7 v and z39 Mu v7 Where these functions are determined by inverting the relation in equation 62 Note that if if 7 16204 06 ilel7 the system is close to a resonance of order k then equation 63 is an example of the general system i em as 64gt Where f R X R X R gt7 R is smooth and of period T gt 0 in t 61 Averaging Regarding equation 6 47 set 7 1 T ame 7 ms fame fz 7 T fyt0 dyl 0 Here f is the mean of f7 and f is the oscillating part of Now set I I y6wy7t767 65 Where w has yet to be determined Equation 65 is an example of a nearidentity transformationl Differ entiating equation 65 and using equation 64 yields that 8w 1 D 39 39 7 7 e ywy z e at 8w fy w fy w7t7 55 Since 11 eDwal ll 7 eDyw C62 27 T Kapitula upon doing a Taylor expansion for f and fone sees that y 7 M Mao 7 27 e2f1ltwgt 0amp3 where f1 m 7 Dyfltytogtwltytogt 7 Dywm t 0W gm 66 If one sets 27 Mao 67 then one nally arrives at the system y39efye2f1yt0esi 68 Note that f1ytT f1 y7 t hence7 equation 68 can be thought of as a periodically forced autonomous systemi Theorem 61 Averaging Theorem Consider equation 64 and the associated averaged system Q e y 69 The following are true a Ifzt is a solution to equation 64 and yt is a solution to equation 69 and iflz0 7 06 then 7 06 on a time scalet N 6 1 b pro is a hyperbolic xed point for equation 69 then for e gt 0 su ciently small equation 64 possesses a unique hyperbolic periodic orbit 75 t Po 06 of the same stability type as po 0 pro is a hyperbolic xedpoint and ifzst E WSWE and yst E Wsp0 then lzs07ys0l 06 implies that 7 06 for all t 2 0 d pro is a hyperbolic xed point and ifzLl t E W 75 and y t E W p0 then lzu0 7 y 06 implies that lzut 7 y 06 for all t S 0 Proof Let ye t denote the solution to equation 6 87 and let yavg t denote the solution to equation 69 If zt is the solution to equation 6 47 then via the transformation in equation 65 one has that lzt7y5tl elwy5tel Oei 610 The estimate follows from the fact that T fy7t7 dt 0A 0 Thus7 upon using the triangle inequality Mi 7 yavgtl S Mi 7 yetl lye yavgtl7 the result of part a is proved if lye t 7 yavg 06 for t N 6 1i Now consider equation 68 and equation 697 and recall that the solution to each is given by an integral equation Letting yd 2 ya 7 yavg L be the Lipschitz constant of f7 and C the maximum value of fl7 and subtracting the two solutions eventually yields 1 mm WON e20teL molds 0 Math 571 Class Notes 28 Applying a generalized Gronwall s inequality see 9 Lemma 412 yields that z MOM S yd0 eeLt 620 eeLtis d5 0 C mm 7 Thus if lyd 06 then one can conclude that lyd 06 for 0 S t S 6L 1 For part b the general idea is to construct a Poincare map for equation 68 and equation 69 and then show that the xed points and manifolds for each are within 06 The change of variables in equation 610 is used implicitly throughout Let H0 H5 denote the timeT Poincare map for equation 69 equation 68 Note that He is within C62 to H0 since T is xed independent of 6 Set 1 9051076 1 2010 p 7 p and note that for each 6 gt 0 one has that go 1006 0 Furthermore since DPHO p0 eETD po one has at 11111 D1790ltp07 6 TD PoL 543 which is nonsingular Now since He is 62close to H0 one has that 11111 9410076 0 lim 13179410076 TD pol eaOt 5H0 Hence upon applying the lmplicit Function Theorem one has a unique curve of points p5 which are within 06 of po that satisfy 95105 e E 0 Furthermore since 105 P0 06 one has that Dynamo eeTWw 08 so that the xed point p5 is also hyperbolic The proof for parts c and d is given in 9 Chapter 41 1 Remark 62 One has that a Parts c and d of the averaging theorem can be paraphrased to say that W5 p0 approximate to 06 the stable and unstable manifolds of the Poincare map of the full system in equation 64 b The persistence of the critical point as a periodic orbit is guaranteed via the lmplicit Function Theorem as long as A 0 0Dfp0 However in the case that ReA 0 one cannot make a conclusion regarding the stability of the xed point for the Poincare map without further analysis Remark 63 In some cases secondorder or even higherorder averaging may be required In such a case upon setting T 132 2 f1ltztogtdt and using the second transformation y I 2 e2wzt 6 one nds that equation 64 becomes 2 e z 8132 09 The secondorder averaged equation is then 2 62f1z One can show that the solutions are a good approximation on a time scale of 06 2 29 T Kapitula Figure 1 The zero set of yew The solid line corresponds to stable solutions While the dashed line corresponds to saddle solutions 611 Example Duffing equation Consider equation 61 With fzz39t ycoswt 7 6139 7 113 and suppose that Aug 7 w2 69 All of the parameters are assumed to be positive Setting k l in equation 63 eventually yields the system 11 coswt 7 vsinwt 7 w6u sinwt vcoswt w aucoswt 7 vsinwt3 7 ycoswt sinwt 611 i Qucoswt7vsinwt7w6usinwtvcoswt w au cos wt 7 v sinwt3 7 ycos wt coswti The averaged equation T 27rw associated With equation 611 is given in polar coordinates by T 276wT 7 ysing w 6 3 3 612 T9 7 QT 7aT 7 70086 i 2w 4 The critical points for equation 61 are given by T 7sint97 969 07 Where 3 ya 9 I 96w2 sint9 1172 sin3 9 6w3 cos 9 Note that T 2 0 necessarily implies that the only relevant critical points satisfy sint9 S 0 Set 3 a 6w 1 lay One solution to 969 0 is given by 1 7r Qsmgsn 1 7 lt gt lt a2 2 Math 571 Class Notes 30 A Taylor expansion of 96 about 95 95 yields the normal form gnaw 7 7am 7 as W 7 emf hence a saddlenode bifurcation occurs at 95 9 Upon using the fact that for gtgt 1 one has that 969 0 for 9 N 0 mod 7r one gets the diagram for the critical points seen in Figure 1 also see 9 Figure 421 The stability types of the branches of the steady solutions are obtained by consideration of the eigenvalues of the linearized averaged equation 61 Since all of the critical points are hyperbolic they persist as periodic orbits for equation 611 Sample phase portraits for equation 61 in the case that 6 f 0 are given in 9 Figure 422 and a comparison of the averaged ow with the Poincare map for equation 611 is given in 9 Figure 423 If one writes equation 61 in Cartesian coordinates then an easy calculation shows t at traceDf 766 lt 0 so that an application of Bendixsonls criterion yields that there exist no periodic orbits If 6 0 the original system is Hamiltonian and the Hamiltonian for the averaged ow is given by 7 3 Huv I 7 9u2v2 7au2v22727u 4w 8 The phase portrait for the averaged system is given in 9 Figure 441 Since the periodic orbits are not hyperbolic one cannot conclude that there exist invariant tori for equation 61 see Theorem 66 This question is left open for further study 612 Example BoseEinstein condensates Recall equation 15 726R2p01 7 p0 sin 2Aq 7am 7 0 7 ail 7 a50gtplttgt 613 6R2 aim 04300 ai11 0 0 3112 0052A 2P0 1 7 be M where pt T pt for some T E R3 Upon setting 1 AM I 1 7 07 1 A iAMt equation 613 can be rewritten as 20 726R2p01 7 p0 sin 21 cos Amt 7 cos 212 sin Amt 7 50 i1 180100 6R2 aim 20 811 40 811 1800 ai11 0 614 18112p0 7 1 sin 21 sin Ant cos 21 cos AMtD Note that under that assumption that pt is even one has that equation 614 is invariant under the actions MM1775 H 307 117 t Po byt H P0711 Ml 615 If one now assumes that 2 12 W R A6 T kAM k E N 616 then the results of Theorem 61 apply to equation 614 Writing T plttgt7pavzalttgt 0 zalttgtdt7o 31 T Kapitula yields that the averaged system is given by 50 0 L p p 2 1 o o o 1 7 6 17 7 EMO H 0 00Pav 56A 04111 20 011 40 011 aooo a111 0 A The dynamics associated with equation 61 are relatively uninteresting and hence it will be considered no longer 7r 4 37r 4 Figure 2 The phase portrait associated with the averaged equation 618 Now consider equation 614 in the event that e 0 and that 6tT 6t for some T E Rf This case arises physically 1n the presence of a Feshbach resonance 22 23 Assuming equation 616 with A 1 one can again apply Theorem 61 to equation 614 For ease now assume that 6t is even so that 00 6t 26k coskAMti 160 Note that this assumption implies that the invariances associated with equation 615 still hold If 60 0 the averaged system is 30 651300 30 5111211 1 65101811 70 i E 0052111 618 Note that if 61 0 then equation 6 18 reduces to the trivial system hence if 61 f 0 the system is being forced at resonance The solutions to equation 6 18 satisfy W17 ol0 31100821 or If 1811 1 then equation 618 would be a Hamiltonian system however an application of the Holder inequality shows that agu minsup qoltzgt2sup l41rl2 E E and as such one generically has that 1811 lt 1 Math 571 Class Notes 32 Remark 64 If 60 y 0 then the term 560 aim 20 811 40 811 1800 allll ol is added to the equation for in equation 618 Thus 60 can be used as an unfolding parameter The critical points satisfying g 6 01 1 E Tr437r4 are hyperbolic and as a consequence persist as periodic orbits for equation 613 The critical points 20 12 1 E 07r2 are nonlinear centers and the eigenvalues of the linearization are given by 1 2 A iilt7a8211gt 5 These critical points also persist as periodic orbits for equation 613 The phase portrait for equation 618 is given in Figure 2 Since the periodic orbits are not hyperbolic one cannot conclude that there exist invariant tori for equation 613 see Theorem 66 This question is left open for further study 62 Local bifurcations goba behavior and Hamiltonian systems One can prove the following theorems via an analysis of the Poincare map as in the proof of Theorem 61 The details can be found in 9 Chapter 434 It should be noted that if a Poincare map undergoes a Hopf bifurcation then that implies that for the map there exists an invariant closed curve 7 however the dynamics of the Poincare map on 7 do not need to be periodic Theorem 65 If equation 69 undergoes a saddlenode or Hopf bifurcation the for e gt 0 and su ciently small the Poincar map associated With equation 64 also undergoes a saddlenode or a Hopf bifurcation Theorem 66 If there exists a hyperbolic periodic orbit 70 for equation 69 then for the Poincar map associated With equation 64 there exists an invariant closed curve 75 Which is close to 70 If the original system equation 64 is Hamiltonian then the transformation leading to equation 68 can be chosen so that the system remains Hamiltonian 8 Chapter 9 In particular the averaged system equation 69 is Hamiltonian If I E R2 the solution curves are the level curves of the averaged Hamiltonian If there exists a homoclinic or heteroclinic orbit for the unperturbed Poincare map the relevant manifolds necessarily intersect in a nontransverse fashion hence we cannot expect it to be preserved for the system equation 68 63 Comparison With a multiple time scales expansion The bulk ofthe material presented herein can be found in 26 Chapter 82 Again consider equation 64 and assume that for some N E N the solution has the form N 167 Zejletn39 71 at 619 By the chain rule one has that g 3 3 dt 6t 667 so that equation 64 can be rewritten as 610 610 611 2 an 912 3 7 2 at eltaT atgte 97 at Oe 7efzoezl0e te 620 Since fIo 611 me f107t70 6 ltDxf107t7011 zot0gt C62 33 T Kapitula upon equating like powers of e in equation 620 one arrives at the sequence of equations 6t 0 811 810 7 7 621 at 97 flt107t70 812 7 811 6f E 7 i Dxf107t70xl l E107t707 which can be solved successivelyi Integrating the rst equation in equation 6 21 yields 102 T 147 and integrating the second equation in equation 6 21 yields L BA 11t739 767 fAess0 dsBT B0 0 622 0 739 In order that terms do not become unbounded we must now enforce the secularity condition that ot li fA53s0 d8 07 ie T g fAs0ds 6 23 0 ie dA a efAi Note that equation 623 is exactly the averaged equation 69 Further note that equation 62 can then be rewritten as 11t739 0 fAess0 7 fAes dsBT B0 0 In order to determine BT one proceeds as above Plugging the solutions for 10 and 11 into the nonhomo geneous equation for 12 in equation 6 21 and applying another secularity condition eventually yields the nonhomogeneous linear system T g DJltABT we no ltz1ltwgte 37 at It can be shown that the expansion in equation 619 where 10 and 11 are prescribed as above yields an C62 approximation on the time scale t N 6 1 26 Theorem 8 21 Remark 67 It is important to note here that the 06 correction to the averaged equation operates on both time scales In particular BT adds an 06 correction on the time scale t N 6 1 and hence its evaluation can be considered to be relatively unimportanti For a simple example consider 139 61 7 12 cos2 t 624 By equation 623 one sees that dA 1 E A17 A A0 A0 iiei A0 A i T A0 17 A0e72 Math 571 Class Notes 34 Now7 upon setting 111157 739 0 fAess0 7 fAes ds 1 dA EO ECOS2SClS 1 t yes2 114017 A00 W cos23 ds7 one has that 11t7 739 u t7 739 B7 7 where satis es the ODE dB 1 1T 6 1 7ABTO 7E 7Acos23 u tq dt7 B001 It can be seen in Figure 3 that u1t739 is oscillatory on the time scale t 917 and is slowly growing on the time scale t 6 1 Figure 3 A graph of both the averaged solution 147 blue curve and the rstorder fast correction u1t 739 red curve for equation 624 Here 5 005 and A0 006 64 Almost periodic vector fields When discussing the Feshbach resonance in Section 61127 in order to apply the theory of averaging we were required to make the somewhat arti cial assumption on the period of the resonant forcing We will now generalize Theorem 61 so that this restriction can be removed A complete discussion of almostperiodic differential equations can be found in 641 Preliminary estimates Consider an almost periodic function pt R i gt R which can be written as 00 W Z pje ki 625 F700 35 T Kapitula where M E R and Taco 1 T 4A z pj hm lTpte 7 dt lf pt were periodic then one would have that M 27raj for some a E R3 Now assume that A0 0 and that M f 0 for j E Z0i Further assume that M Ak for j kl Under this assumption note that the average is given by 1 T 100 Tlggo 7T POth Now set 1 T 2 2 p 7 hm i pt dt ll 2T iTl Taco 00 Z lpjl27 F700 and note that termbyterm differentiation yields 00 WOW 2 WW 1700 We now have the following preliminary estimate which is useful in determining if an almostperiodic function is uniformly bounded Lemma 68 If 1 Z P lt 00 6 26 0 1 then for some C E RJF sup MOW S Cllpll llPlll 16R Proof First note that lpol2 S Upon using the representation of pt one then sees that 1 lle S lmHZmel 0 J 12 12 1 00 S HPH Z A lpjl2 j 0 J 1700 The second line follows from the CauchySchwarz inequalityi D Now consider gm 2 ltpltsgt wads Pj n z 10139 1 me 7 7 i 0 0 90 Zgjen ty 0 where the coef cients 92 are implicitly de ned If one assumes the bound presented in equation 626 then an application of the CauchySchwarz inequality yields that 12 12 1 wow 2 Z W SCHPH 0 139 1700 Math 571 Class Notes 36 hence7 the average of gt is wellde ned Similarly7 if one assumes that there is an e gt 0 such that MA 2 e for all j E Z07 one sees that l V 2 7 V 2 Zlgjl S 6 lejl 0 0 If one again assumes equation 6267 then the existence of such a lower bound is implied Combining these results yields Lemma 69 Suppose that the exponents satisfy the bound in equation 626 Then z glttgt ltpltsgt wads 0 is an almost periodic function With a representation as given in equation 625 Furthermore there is a C E R such that llyll S Cllpll 642 The averaging theorem Let us now consider equation 64 under the assumption that fzte fz7 t7 and fzt Za z cosjt 1211 sinjt7 Aj E R 627 j1 Where T l I T211100 TO fzt dt 628 Thus f is assumed to be almostperiodic in t E RJF and that it is being Written using the notation of realvalued functions It Will further be assumed that M Ak for j k and that equation 626 holds true De ne the local average fT off by T Mm fztsds and note that if f is Tperiodic7 then fT f 267 Lemma 323 In fact7 we have more Lemma 610 Under the above assumptions on fzt one has that fT17t 1 01 Proof One has that 1 T Mam fltzgt T Wm e m as 0 1 tT 7 T ms e m as 2 By Lemma 69 one has that fzt 7 is almostperiodic7 and hence uniformly bounded7 so that lfT17t l S C D If one considers a Lipschitz continuous map 45 R gt gt R With Lipschitz constant L7 then one has that 1 T W 7 mm s T W e lttsl as 0 ltlTLd lt1LT 7T0 8872 37 T Kapitula hence7 W Tt 0T 629 We now have the following preliminary estimate Lemma 611 Consider equation 64 If 10 fltzltsgtsgtds then t M fTltzltsgtsgtds M 0 fort 06 1 Proof Set T 5 t T R12 1 mom was 122 1 1mm sgtm 8 7 fr8w sgt1drds T 0 0 T 0 0 By de nition7 and after some algebraic manipulation7 one has that z Tt fTzss d3 R1 R2 0 Now let L be the Lipschitz constant associated with f7 and set MI sup sup erogsze An easy estimate shows that 1 T 5 1 lt 7 lt 1 11311 TA 0 Mdrds 2MT while L t T lel S 7 lzrs7zsldrds T 0 0 L z T 7 5 eT lfzyyl dydms 0 0 s L T 1 S 67 MT de3 ieLMTt T 0 0 2 The result now follows 1 We can now prove the quasiperiodic averaging theorem Theorem 612 Quasiperiodic Averaging Theorem Consider equation 64 Where fzt is given as in equation 627 Let yt be the solution to the averaged system Q e y y0 900 Where f is de ned in equation 628 One then has that 9W Mt 0612 on the time scalet 06 12 Math 571 Class Notes 38 Proof The solution to equation 64 is given by m 10 60 fzss ass at z fltzltsgt75gtds By equation 629 one has that 4W Tt WT and by Lemma 611 one has that M thltzltsgt73gtds0ltTgtl Hence7 t W fTltzltsgt78gtds0ltTx so that t 1t 10 60 fTzss ds 06Tl Now7 the solution to the averaged equation is m mm e Ems as so that l1t WM S 601 lfTIS75 fy8l 018 06T S EOtlfT1878 fTy878l 018 601 lfTy878 fy8l 018 06TA As an application of Lemma 610 one then has that Wt ytl S 601 lfTIS75 fT 373l 018 0a 06TA Using the Lipschitz continuity of fT and Gronwallls inequality then yields that Wt WM 0a 06Tle L A Assuming that tT 06 12 nally gives the desired result D It is clear that the result of Theorem 61 is not as strong as that of Theorem Gill The necessary re nement of Theorem 61 is given in 267 Lemma 376 and Will be stated below as an unproven corollaryl Corollary 613 Consider equation 64 Where fzt is given as in equation 627 Let yt be the solution to the averaged system 7 y 6fy7 y0 900 Where f is de ned in equation 628 One then has that IO Mt 06 on the time scalet 06 1 Remark 614 The theory presented above is far from complete A more recent article Which discusses both rst and secondorder averaging in the quasiperiodic case is 39 T Kapitula 643 Example BoseEinstein condensates Recall equation 6 14 20 726R2p0l 7 p0 sin 21 cos Ant 7 cos 212 sin Ant 7139 50 i1 180100 6R2 aim 20 811 40 811 1300 ai11 0 630 18112p0 7 1 sin 21 sin Ant cos 21 cos Autl and as in the case of the Feshbach resonance assume that e 0 and that 6t T 6t for some T E Rf Again assuming that 6t is even With zero average one has that 00 km 6t 7 26k cos a t a E Rl 161 Note that A a iTl 27f We Wish to apply Corollary 613 to equation 630 The exponents associated With the quasiperiodic expansion of the vector eld are given AjAplt1 gt jez Hence equation 626 is satis ed if a f j for j E No This condition is automatically satis ed if T is not integrally related to the natural frequency The case that T is an integral multiple of the natural frequency has already been considered in Section 61in Under the above nonresonance condition the averaged equations then become ie the dynamics are trivial 65 Subharmonic orbits As in Section 543 consider 9395 1 657W 631 Where f R2 gt7 R2 is a smooth Hamiltonian vector eld and g R2 X R gt gt R2 is smooth With gzt T gzt for some T gt 0 For the case of simplicity assume that for z uvT and f f1 f2T that 8H 8H f1 7 my f2 7 E for a smooth Hamiltonian Hu 1 Assume that When 6 a there exists hyperbolic critical point p0 Which is a saddle point b there exists an orbit qo t homoclinic to p0 Let To Z q0t t E R U p0 Further assume that a the interior of F0 is lled With a continuous family of periodic orbits qa t a E 71 0 furthermore i one sets d F I 39 f 7 z o lz at then we have that hm sup data m 7 0 0470 16R Math 571 Class Notes 40 b if Ta is the period of gut and ha I Hqat then Taha is differentiable with dTa E gt 0 Note that the above two assumptions imply that TO A 00 monotonically as a A 0quot Now let Ta m n where m E N an n E N are relatively prime If n l the orbit is known as a subharmonic orbit and if n 2 2 the orbit is known as an ultrasubharmonic orbitl Note that got is also mTperiodic for any n E N If pa I get to for some to E 0 Ta then for the unperturbed Poincare map H0 one has the periodic orbit pm new 7H 1pa7 ilel H6 pa pal Thus we can consider the persistence and stability of these orbits for the perturbed probleml We will now use a transformation on equation 631 which will make the problem more amenable to analysis As is seen in 8 Chapter 103 there exists an invertible symplectic transformation to actionangle variables within the interior of To given by IIuv 99uv with inverse uUI9 vVI9l The transformation becomes singular at For The coordinates I and 9 are nonlinear polar coordinates and are chosen in such a way that for the unperturbed problem one has that I 0 with 9 E 027rl Under this transformation the unperturbed Hamiltonian becomes HUV E HI and equation 631 can be rewritten as I e ltggl 92 I 6FIt9t u v 97 6H 6 9 6 9 7 91 01 9 t 632 fistf agl l EQZ A7 5 77 The new ow is depicted in Figure 4 The quantity 9U is the angular frequency of the closed orbit with action I and energy lf qa has period Ta and action Ia then one has that Qua 27rTw Since dT 7 dT dh gt 0 W 7 E3 one has that d9 2 dT 7r 3 fig lt 0 633 for all a 6 710 ow choose a resonant orbit with period mTn and action Imml Upon setting 9mm 2 9Imm and 6 I 612 perturb from this orbit using the rotation transformation I Imm 6h 9 9mm 15 6 34 Substituting equation 634 into equation 63 then yields that h 6F 17 an t 0 62 ab 6 35 15 SQImm z C62 Now by the chain rule one has that BI 6H 7 1 BI 6H 1 a w h a BHE 5137 41 T Kapitula Figure 4 The phase portraits associated With equation 631 left and equation 632 right he shaded area represents the region governed by equation 6 so that Pam amt ab t Using the notation of Section 31 set mT Mm s dltfltqalttgtgtgltqalttgttsgtgtdt 636 0 The function Mmn is known as the subharmonic Melnikov function see 9 Chapter 46 for an alternate derivation Note that as a consequence of got being mT periodic one has that Mmns mT Mmws 637 Assuming that F and 9 are bounded Theorem 61 can be applied to equation 635 to get the averaged system A LMmn L 27m 9m n 638 q so1mm It is important to note that equation 638 is a Hamiltonian system With Hamiltonian 7 L mn L 1 Hhq576lt27mM dab 29Imnh2gt hence the solution structure to equation 638 is completely known Furthermore as a consequence of equation 637 one has that the vector eld is maximally 2mrperiodic in et the zeros of Mm be denoted by 31 8k 6 0mT If the zeros are simple then as a consequence of the periodicity of Mmn one has that k 2E for some Z 6 No The critical points for equation 638 are then given by hwy 05j mm j 1k As a consequence of equation 633 one has that the critical point is a saddle if Mm y lt 0 and a nonlinear center if Mm y gt 0 Note that if the zeros of Mm are simple then there is an even number of saddle points and nonlinear centers The Averaging Theorem 61 then implies that there saddletype orbits to equation 631 near the saddle points of equation 638 and periodic orbits near the centers Whose stability types are not determined by this 06 truncation Since the unperturbed system is Hamiltonian one has that the associated Poincare map H0 is area preserving7 and hence detDH0 1 Now suppose that the perturbation is uniformly damping7 and as a consequence satis es the condition that V g lt 0 Since the perturbed threedimensional flow contracts volumes like ew39g 7 the perturbed Poincare map He must satisfy detDll lt 1 Since the eigenvalues M E 0DH5 satisfy AlAg detDll5 one then necessarily has that Math 571 Class Notes 42 a all periodic points of 1 15 are sinks or saddles b there exist no simple invariant closed curves The second conclusion follows from the fact that the interior of a simple invariant closed curve is reduced in area under an application of 1151 It is clear that an analogous result holds if V g gt 07 With sink being replaced by saddle Remark 615 The interesting case that V g E 07 Which implies that He is an area preserving map7 Will be discussed in Section 6161 651 Example Consider the system 11 v17 112 v22 eau 7 uu2 v22 u cos t 639 v 7u17 u 1 eav7vu 1 Without loss of generality it can be assumed that B E R3 The system has the feature that When 6 0 all circles are invariant curves7 and the circle of radius one is composed solely of critical points The Hamiltonian associated With the unperturbed problem is given by 1 2 2 1 2 2 2 Huv 1 7 1 1 Under the transformation to actionangle variables7 u I VZI sint 7 v I VQI cost 7 equation 6139 becomes I e 2aI74I22 Isin2t9cost T B l 6140 9 17 2I e singcosgcostl The unperturbed Hamiltonian is now given by H I I 7 I 2 and the period of the unperturbed orbits is 27f T I 71 1 7 2I Let us study perturbations of the resonant orbit of period 47f With action I 141 As in equation 6347 make the transformation 1 1 I 16h 9 t 6 512 After some trigonometrical expansion one sees that the transformed equation is given by 1 1 1 h 16 2a 71 ltcost Esin2q5sint7 1cos2tcos2 gt 1 1 62 2a 72 ltcost Esin2q5sint7 51cos2tcos2 gt h 463 6141 45 726h 62 cos 245 sin 2t 1 cos 2t sin 245 Note that7 unlike equation 61357 the explicit form of the C62 terms in equation 6141 are known 43 T Kapitula In order to average equation 6 417 use the transformations given in equation 65 and equation 67 to get 1 1 1 h gt gt h 166 ltsint 7 Zsin2 cos2t 7 1cos2q5sin2tgt 7 ab gt gt After a great deal of calculation it is seen that the averaged equations up to C63 are given by 1 1 h 16 2a 717 E costh 262 a 717i c0s2 063 6 42 ab 726 i62 sin2 063 I believe that there is an error in 9 equation 41718 Note that equation 64 can be considered on the torus h7 E R X 07 674a2 Figure 5 The phase portraits associated With the 96 averaged equation 642 The Hamiltonian associated With the 06 terms in equation 64 is given by 7 1 1 2 Hhq I 16 2a 71q 7 l sln2 4h 1 Note that if a 127 then H is 7r periodic in 45 hence7 one may expect this line to be a bifurcation line in parameter space 5 gt 212a 7 1 Math 571 Class Notes 44 then there will exist two critical points for 45 E 07r otherwise there will exist none The critical point satisfying sin 245 lt 0 will be a saddle point and the other will be a nonlinear center Upon reviewing equation 639 it is seen that Vg 2a costi4u2v2 hence if B lt 2lal and a E R the nonlinear center will become a sink at C62 Note however that no critical points exist which satisfy this condition Sample phase portraits at 06 are given in Figure 5 The interested student should consult 9 Figure 472 to see the phase portraits at C62 It is an interesting exercise to fully develop the bifurcation diagram 66 KAM and Twist theorems As a consequence of the discussion in Section 65 the leading order effect of resonant perturbations is now understood In this section we will undertake the study of those orbits which are not forced at resonance 661 Algebraic preliminaries The following material can be found 10 Let 9 E R be such that it is a root of a polynomial equation with integral coef cients Such a number 9 is said to be an algebraic number lf 9 satis es an equation of degree n but none of lower degree then it is said to be an algebraic number of degree n Set An I 6 9 is an algebraic number of degree It is easy to see that 9 6 A1 implies that 9 E In fact A1 Q so that A1 is a countable set It is also easy to see that if q 6 A1 then ilqlln E An for each n 2 2 It can be shown that An is countable for each n 2 1 hence the set AI 0 Am 711 ie the set of algebraic numbers is itself countable and has Lebesgue measure zero One has that 9 is transcendental if 9 A As a consequence almost all real numbers are transcendental Algebraic numbers have the property that they cannot be approximated too rapidly77 by rational num bers A number 9 is said to be approximated to degree k if the inequality 17 7k l7 7 el lt lql 4 has an in nite number of solutions 104 6 Z It can be shown that if 9 E Q then 9 can be approximated to order one and to no higher order Furthermore if 9 g Q then 9 can be approximated to order two The following theorem which is due to Liouville gives a characterization as to how well the algebraic numbers can be approximated Theorem 616 Ift E An for some n E N then 9 cannot be approximated to any order greater than n Remark 617 Two direct consequences of Liouvillels theorem are a if 9 E An then there exists an e gt 0 such that 379l 26M pyqez 643 b if 9 A7 then it can be approximated by rational numbers arbitrarily quickly 45 T Kapitula 662 KAM and Twist theorems Regarding equation 6317 suppose now that the ow is volumepreserving7 ie7 V g E 0 Once again consider the transformed equation 632 and note that the unperturbed problem is 139 0 9 91 644 where Qua 27rTa The Poincare map associated with equation 644 is given by HM 6 16 9W 645 192WTZD Since Q U 0 recall that Q U lt 0 the map H0 is known as a twist map Note that it preserves area Finally note that if TTa E Q then the associated orbit is periodic otherwise it densely lls the circle I Ia The perturbed mapping is given by H5079 16f179791T691797 646 where f and g are bounded and 27rperiodic in 9 Since the ow is volumepreserving7 the map 1 15 is area preserving As a side remark since the perturbed map is areapreserving one has that any elliptic points are actually centers One has the following important result concerning the existence of tori Figure 6 The Poincare map associated with equation 646 The green curves are invariant curves whose existence is guaranteed by Theorem 619 The xed points7 and their stability type are guaranteed by the discussion in Section 65 Theorem 618 If Q U 0 and e gt 0 is su ciently small then He has a set of invariant closed curves ofpositive Lebesgue measure close to the set I Ia The surviving closed curves are lled With dense irrational orbits Unfortunately Theorem 618 does not yield any information as to which of the unperturbed closed curves persist As a consequence of the discussion in Section 65 we know that this set does not include any unperturbed curve which satis es TTa 6 The following result lls in this gap Theorem 619 Twist Theorem Under the assumptions of Theorem 618 if TTa E An for some n 2 2 then the invariant curve a persists as an invariant closed curve for equation 646 Furthermore the curve is lled With dense irrational orbits Math 571 Class Notes 46 The persistence of the invariant closed curves has an important implication regarding the behavior of the mapping Her Let 71 and 72 be two such invariant curves and assume that 71 C int72l Set the annulus A5 I ext71 int72 and suppose that p 6 A5 see Figure 6 Since the map H5 is induced by the ow of equation 631 as a consequence of uniqueness of solutions to equation 631 one must that Ha z E ii In other words Ugh45 C Aer Since Q U 0 there will exist an initially resonant orbit within Aer As a consequence of the discussion in Section 65 one knows that this resonant orbit breaks into an even number of xed points half of which are centers and half of which are saddle points The invariant manifolds of the saddle points are contained within A5 and they must intersect otherwise He would not preserve area In general one can expect that some of these intersections will be transversel In this case the result of Theorem 51 is relevant and consequently one has the existence of homoclinic tangles and all of the attendant complicated dynamicsl 67 Example Bose Einstein condensates Assuming that there is no magnetic trap and after several rescalings the steadystate problem associated with equation 11 in the repulsive case can be written as Lj q 7 L13 ECOSHI 647 see 25 for the details The periodic solutions to equation 64 are given by lt k W lt k eix gr 7 1k2snz ziim where k 6 01 and satisfy the initial condition 16 WW 07 4077 Vim The period of qz k is Tk I 41 k2Kk where KUs is the elliptic integral of the rst kind ilel 7r2 1 1m d l 0 l 7 k2 sin2 15 The period has the property that Tk gt 0 with 6133 TUs 2 16131117 TUs 00 Finally 41 16 is odd in 1 Upon following the discussion in Section 65 one has that for the unperturbed problem the action is given by Tk mg I qzywdz W lt1k2gtEltkgt 7 1 713mm where Ek is the elliptic integral of the second kind ilel EkO7r2 1ik2sin2qbdqbl 47 T Kapitula The evaluation of the integral follows from the identities given in 24 Chapter 410 The associated angle for the unperturbed problem is given by 27f 91 k21 90 The spacedependent Hamiltonian associated with equation 64 is given by l l l 1 771017471 1 42 42 144 542 0050 Although it will not be done herein as it is not necessary for the subsequent analysis one can now compute the Hamiltonian in actionangle variables see 25 equation Remark 620 Although it will not be pursued herein this exact evaluation of the Hamiltonian in action angle variables allows one to easily apply a higher order perturbation theory This avenue of attack is pursued in 25 As seen in Section 65 the important quantity to calculate is the Melnikov function given in equa tion 636 One has a resonance if for mn E N 7 m 7r 1 n 2Kk 41 k For ease now assume that n 1 Upon using the de nition in equation 636 the restriction of equa tion 648 and the identity 6 48 snu k2 cnu k2 1 one sees that the Melnikov function associated with equation 64 is given by m 397 k2 m7r M if immMc c n s1nmq 649 where 2mKk 2 m Mckm I 2mKk cnu 16 cos lt2Kltkgt ugt dui 6 50 Figure 7 A plot of the constant MCUC 771 de ned in equation 650 for m 2 4 A plot of the constant M006 m for m 2 4 is given in Figure 7 First as a consequence of the fact that the Fourier cosine series for cnu k2 is given by mm k COOS 2 CW COS 212 Math 571 Class Notes 48 25 equation 52 one has that MC h2 1 E 0 for Z 6 N0 Thus if m 2E 1 the theory presented in Section 65 does not give a de nitive answer and a higherorder Melnikov theory is necessary see 24 Chapter 411 for some details As a consequence it will henceforth be assumed that m 2A for some Z 6 N this assumption is also taken in 25 As a side remark since cnu h2 is holomorphic on a strip containing the real axis one has that for xed k 6 01 and any N E N I 13100 Mck 2i 201V 0 Hence one must also have that the upper bound on e for which the theory presented in Section 65 is applicable approaches zero as Z A 00 The primary point of 25 is to resolve this limit In any event one nally gets that 7r 2 Z 1 k2 1mg Appealing to equation 638 and the subsequent discussion one sees that the critical points for the averaged equation in the resonance band are given by M21015 Mck 2i sinegas n on 7r n0471 Thus the resonance band has 2E centers and 2K saddles and the centers correspond to those points in which n is odd and the centers correspond to those points for which n is even Remark 621 The physical interpretation of this result is that as the frequency of the optical lattice increases the number of critical points within a particular resonance band increases Thus increasing the frequency of the optical lattice increases the complexity of the dynamics for the steadystate problem REFERENCES 1 J Carr Applications of Center Manifold Theory SpringerVerlag New York 1981 2 J Ellison and HJ Shih The method of averaging in beam dynamics In Y Yan and M Syphers editors AIP Conference Proceedings 326 Am lnst Phys 1995 3 J Evans Nerve axon equations 1 Linear approximations Indiana U Math J 2128777955 1972 4 J Evans Nerve axon equations 11 Stability at rest Indiana U Math J 22275790 1972 5 J Evans Nerve axon equations 111 Stability of the nerve impulse Indiana U Math J 2225777594 1972 6 J Evans Nerve axon equations TV The stable and unstable impulse Indiana U Math J 242116971190 1975 7 A Fink Almost Periodic Di erential Equations Lecture Notes in Mathematics 377 SpringerVerlag 1974 8 H Goldstein Classical Mechanics AddisonWesley 2nd edition 1980 9 J Guckenheimer and P Holmes Nonlinear Oscillations Dynamical Systems and Bifurcations of Vector Fields volume 42 of Applied Mathematical Sciences SpringerVerlag 1993 10 G Hardy and E Wright The Theory of Numbers Oxford University Press London 1956 11 CKRT Jones Stability of the travelling wave solutions of the FitzhughNagumo system Trans AMS 286 2431469 1984 12 CKRT Jones Geometric singular perturbation theory In R Johnson editor Lecture Notes in Mathematics 1609 SpringerVerlag New York 1995 13 CKRT Jones T Kapitula and J Powell Nearly real fronts in a GinzburgLandau equation Proc Roy Soc Edin 116Az1937206 1990 14 T Kapitula Singular heteroclinic orbits for degenerate modulation equations Physica D 821amp23659 1995 49 T Kapitula 15 16 17 18 19 20 21 22 23 24 25 26 27 28 T Kapitula Existence and stability of singular heteroclinic orbits for the GinzburgLandau equation Nonlinr earity 936697686 1996 T Kapitula Bifurcating bright and dark solitary Waves for the perturbed cubicquintic nonlinear Schrodinger equation Proc Roy Soc Edinburgh 1281425857629 1998 T Kapitula Stability criterion for bright solitary Waves of the perturbed cubicquintic Schrodinger equation Physica D 1161295420 1998 T Kapitula and P KeVrekidis BoseEinstein condensates in the presence of a magnetic trap and optical lattice submitted T Kapitula and P KeVrekidis BoseEinstein condensates in the presence of a magnetic trap and optical lattice twomode approximation submitte T Kapitula and S MaierPaape Spatial dynamics of time periodic solutions for the GinzburgLandau equation Z angew Math Phys 4722657305 1996 P KeVrekidis and D Frantzeskakis Pattern forming dynamical instabilities of BoseEinstein condensates Modern Physics Letters B 1821737202 2004 P KeVrekidis G Theocharis D Frantzeskakis and B Malomed Feshbach resonance management for Bose Einstein condensates Phys Rev Lett 9023230401 2003 D PelinoVsky P KeVrekidis and D Frantzeskakis Averaging for solitons With nonlinearity management Phys Rev Lett 9124240201 2003 L Perko Di erential Equations and Dynamical Systems SpringerVerlag 2nd edition 1996 M Porter and P CVitanoVic A perturbatiVe analysis of modulated amplitude Waves in BoseEinstein conden sates Chaos 1437397755 2004 J Sanders and F Verhulst Averaging Methods in Nonlinear Dynamical Systems Volume 59 of Applied Mathei matical Sciences SpringerVerlag 1 J Smoller Shock Waves and Reaction Di usion Equations SpringerVerlag New York 1983 S Wiggins Introduction to Applied Nonlinear Dynamical Systems and Chaos Volume 2 of Texts in Applied Mathematics SpringerVerlag 2nd edition 2003

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