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# Class Note for MATH 3331 with Professor He at UH

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This 20 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 19 views.

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Date Created: 02/06/15

Lecture 25 x 2 Genmztnc ntEvvvetatmn nvsmutmns x 3 QuahtaweAna Ws J W H 2 Wm M WM WW M mm quotA iivlavamZXmm Section 82 t 3 De nitions In Class Exercises Autonomous system and Phase Space Plot Autonomous system Rn phase space 353 n 2 phase plane Trajectory Curve xt t E I in R RHS doesn t depend explicitly I interval on which xt on t is defined X fX X 331 For any 15 xt E Rquot xt solution gt Xt to solution same trajectory o Tangent vectors x t f WM 0 Vector field X gt fX trajectories don t intersect If existence and uniqueness Iill Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 2 2O mam Manama mm D warms s we Mm A 39u um y am a memoquot m H WW 5 2 e RHSs I 27 g dun idepen i quotWWW 039 PM NEW aphmuyunt PM a MM my VON9 39xf yf xf yf a Vsdm he d y 39X y gtlt y Section 82 if Definitions n Class Exercises Predator Prey System Example LotkaVolterra s predatorprey equations R a bFR 1 F c dRF a b c d gt O R number of rabbits F number of foxes I Pa ra m ete rs 5039 trajectory in RF plane a b 2 quotquot 40 with tangent vectors C Z d 30 IC R0 40 F0 20 Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 Section 82 ri Definitions ln Class Exercises Predator Prey System cont Com posite Gra ph Several K 20 4O 6O 80 100 120 140 Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 5 20 Section 82 y O1y 370 0 y0 2 time DIO CS 39 trajectory and Ex y y X 01yl vector field 2 quotquotquotquotquotquotquotquot quotquotquotquotquotquotquotquotquot x 15 ml I y Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 6 20 Section 82 Exercise 821 39Ex 821 Plot i x1tm2t and ii the curve t gt x1tx2t for Xt 26 6 75 6 Le 5310 2 t e t 30205 64 Matlab commands tlinspace02100X12eXptexptX2eXpt figure1plottx1 k tX2 k Xlabel t ylabel X1 and X2 figure2plotX1X2 k Xlabel x1 ylabel x2 axis0 15 O 1 Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 7 20 ln Class Exercises Section 82 Definitions Exercise 823 x1 and x2 Ex 823 Same as Ex 821 for Xt costsintT ie 9011 2 cost 3205 sint solid x t dashed x2t 1 1 1 08 08 06 06 04 04 02 02 N O O gtlt o2 o2 o4 o4 0I6 O6 08 O8 1 1 39 o 5 10 15 20 1 0 5 3 O 5 1 t 1 He University of Houston April 6 2009 8 20 Math 3331 Section 19470 Lecture 25 Section 82 g i Exercise 8222 Plot t gt Xtyt in the xy plane xt 02 yt Initially x0 O and y0 2 then 2 decays as x increases thereafter both x and y oscillate as they decay toward zero H Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 9 2O Section 82 Exercise 8223 Plot t gt Xtyt in the xy plane 260 02 ya 25937 I I II II II II I I I I I 39 I I I I I I I I I I 39 I I I I I I I I I I I I I I I I I I I 00 39 i I I I I I I I I I l I I I I I I I I I I I I I I I I I I I A I I I I I I V x I I I I I 2 S I II II II II II Initially x0 0 and y0 2 Shortly thereafter 2 5 V y decays as x increases Soon both x and y begin a 39 seemingly periodic motion H Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 10 20 Section 82 quot Exercise 8224 Plot t gt Xtyt in the xy plane n x 3 0 3M A At rst x oscillates mildly about 1 while y oscil lates mildly about zero This would indicate a turn 3 ing about 1 0 in the phase plane The oscillations grow larger until both x and y shoot off to oo One I possible solution follows Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 11 2O Section 82 t Definitions ln Class Exercises Exercise 8225 Plot t gt Xtyt in the xy plane A x no 3 A y 0239 0 x 8 00 V Initially x0 0 and y0 2 Thereafter x increases rapidly then decays asymptotically in an 39 3 oscillatory manner to about 5 or 6 Meanwhile y de cays eventually oscillating about zero One possible solution follows Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 12 20 Definitions ln Class Exercises Section 82 Exercise 8217 Ex 8217 Plot i solutions actyt of IVP as functions of 1 ii trajectory IVP 13quot 6m 10y y39 51 I 4y 960 2 5 1 Use ppane6 x 6X10y y 5x y X x 6X10yy 5x4y quotquot y Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 1320 g H a bFR c l dRF 1 Equilibrium points R F O a bFR 0 2 c l dRF 0 Solutions R7 FiT 001T R7 FiT cd abiT Equilibrium points gt constant solutions of ODEsystem RUL FtT Cdi abiT Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 Rnullcline R 0 gt ROandFab Fnullcline F O gt FOandRcd Equilibrium points are intersections of nulcines 119 April 6 2009 14 20 Section 83 ouilibrium Points In Class Exercises Example I several solutions nullclines a 1 a ya and equilibrium points EX1 3 Z 4 2a 7yy using oplane6 x 1Xyx y 42X7yy 39m nullclines 1 0 a y 1 y nullclines y 0 295 I 7y 2 4 Equlibrium points 00 047 10 3525 Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 15 20 In Class Exercises Section 83 cuilibrium Points Exercise 831 Equilibria Jiwen He University of Houston Ex 831 Plot i nullclines and ii equilibrium points for O2a 004ry 53 20 330andy25 Nullcllnes O1y0005y y 0 2 y0andx20 x 02x004xy y 01y0005xy 0 0T 20 SF Use ppane6 Math 3331 Section 19470 Lecture 25 April 6 2009 16 20 Section 83 Euilibrium Points In Class Exercises Exercise 832 Ex 832 Plot i nullclines and ii equilibrium points for 13 4w 2332 90 2 g Nullclines O mOand 2my4 4y ij 2y y O gt y Oandm l 2y4 x 4x2x2xy y 4yxy2 y2 X O 00 1 20quot gt Equullbrla MBA3 072 ppane6 Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 1739 20 Section 83 Exercise 8373 V l EX 837 Consider 73 1 9 SINCE C053 y COSx y l smx a Show that xt t yt sint is solution 1 y Sinx C0590 x 1 1 sint sintcost 1 OK COSSC yISinac y COSt cost sintsintcost OK April 6 2009 18 20 Math 3331 Section 19470 Lecture 25 Jiwen He University of Houston Section 83 uuilibrium Points In Class Exercises Exercise 837b Ex 837 Consider 1 9 S39n37 C0555 33 y COSm y l sinaz b Plot solutions x 1 y sinxcosx y cosx ysinx Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 19 2 0 i r quot 39 Section 83 Euilibrium Points IiiClass Exercises Exercise 837 Ex 837 Consider 33 1 9 S39n5 f C0533 y COSCIj yISlna c Show that yt lt sinat for all t if 300 2 7T2 y0 0 Solution of a satisfies y sin 33 Trajectories don t cross gt yt lt sin at if yO lt sin 5100 Jiwen He University of Houston Math 3331 Section 19470 Lecture 25 April 6 2009 2020

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