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Nonlinear Dynamics

by: Kylie Bartoletti DVM

Nonlinear Dynamics PHYS 7224

Kylie Bartoletti DVM

GPA 3.67

Predrag Cvitanovic

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Predrag Cvitanovic
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This 0 page Class Notes was uploaded by Kylie Bartoletti DVM on Monday November 2, 2015. The Class Notes belongs to PHYS 7224 at Georgia Institute of Technology - Main Campus taught by Predrag Cvitanovic in Fall. Since its upload, it has received 19 views. For similar materials see /class/234272/phys-7224-georgia-institute-of-technology-main-campus in Physics 2 at Georgia Institute of Technology - Main Campus.


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Date Created: 11/02/15
1200 References Chapter 13 Counting Solution 131 A transition matrix for 3 disk pinball a As the disk is convex the transition to itselfis forbidden Therefore the Markov diagram is with the corresponding transition matrix 0 1 1 391 1 0 1 i 1 1 0 Note that 3911 2 T 2 Suppose that T an39 I then Tn1 only bnT an bn r Zuni So an1 an bn bn1 2a with al 1 b1 0 b From a we have an1 an 2on4 Suppose that an olt A Then A2 A 2 Solving this equation and using the initial condition for n 1 we obtain the general formula 1 n 71 an go 1 2 n71 n bn g2 lt71gtgt1 c 3911 has eigenvalue 2 and 71 degeneracy 2 So the topological entropy is 1n2 the same as in the case ofthe binary symbolic dynamics Yueheng Lan Solution 132 Sum of A17 is like a trace Suppose that Aqbk Akqbk where Ak qbk are eigenvalues and eigenvectors respectively Expressing the vector v 11 1 in terms ofthe eigenvectors qbk ie v Ekqubk we Tn Ei A LLj v Anv Ekv Anqubk EkdkA v qbk EkckAZ where ck v qbkdk are constants a As tr A 21M it is easy to see that both tr A and Tn are dominated by the largest eigenvalue A0 That Is initrAni nln1A011nEk H1 asn w lnirni nlnMoiln12kdkni soluCount r 8oct2003 boyscout Versionll 9 2 Aug 21 2007 References 1201 b The nonleading eigenvalues do not need to be distinct as the ratio in a is controlled by the largest eigenvalues only Yueheng Lan Solution 134 Transition matrix and cycle counting a According to the definition of39ll ij the transition matrix is His b All walks oflength three 000000010010010001011000 10011010 four sym bols with weights aaaaacacbcba cbcbaabacbcb Let39s calculate 39ll T3 7 a3 Zabc aQC 1202 T a2 2 abc There are altogether 8 terms corresponding exactly to the terms in all the walks c Let 395 look at the following equality T2 Ek1k2k71Tik1Tk1kg Emil Every term in the sum is a possible path from 239 to j though the weight could be zero The summation is over all possible intermediate points n 7 1 ofthem So T2 gives the total weight probability or number of all the walks from 239 toj in n steps d We take a b c 1 tojust count the number ofpossible walks in n steps This is the crudest description of the dynamics Taking abc as transition probabilities would give a more detailed description The eigenvlues of39ll is 1 i so we get N n 0lt 02 e The topological entropy is then In Yueheng Lan Solution 136 Golden meanquot pruned map It is easy to write the transition matrix Hm The eigenvalues are 1 i The number of periodic orbits of length n is the trace 1T71x3 173gt 2 Yueheng Lan Solution 135 3 disk prime cycle counting The formula for arbitrary length cycles is derived in sect 134 boyscout Versionll 9 2 Aug 21 2007 soluCount 8oct2003 1202 References Solution 1344 Alphabet 01 prune 1000 c00100c c01100c step 1 1000 prunes all cycles with a 000 subsequence with the exception of the xed point 0quot hence we factor out 1 7 to explicitly and prune 000 from the rest Physically this means that 0 is an isolated xed point no cycle stays in its vicinity for more than 2 iterations In the notation of exercise 1318 the alphabet is 1 2 3 0 and the remaining pruning rules have to be rewritten in terms ofsymbols 210 3100 step 2 alphabet 1 2 3 0 prune 33 213313 Physically the 3 cycle 3 m is pruned and no long cycles stay close enough to it for a single 100 repeat As in exercise 137 prohibition of 33 is implemented by dropping the symbol quot3 and extending the alphabet by the allowed blocks 13 23 step 3 alphabet 1 2 13 23 0 prune 213 2313 1313 where 13 j 23 are now used as single letters Pruning ofthe repetitions the 4 cycle i 1100 is pruned yields the to H Result alphabet 1 2 E 113 0 unrestricted 4 ary dynamics The other remaining possible blocks 2173 2313 re forbidden by the rules of step 3 The topological zeta function is given by 1M 1 i 7501 i 751 i 752 i 7523 i 75113 844 for unrestricted 4 letter alphabet 1 2 B Solution 138 Spectrum of the golden meanquot pruned map 1 The idea is that with the redefinition 2 10 the alphabet 12 is unrestricted binary and due to the piecewise linearity ofthe map the stability weights factor in a way similar to 1610 N As in 1710 the spectral determinant for the Perron Frobenius operator takes form 1712 1 1 gm det172 7 i lt17 ock lt1 WW The mapping is piecewise linear so the form of the topological zeta function worked out in 1316 already suggests the form of the answer The alphabet 12 is unrestricted binary so the dynamical zeta functions receive contribu tions only from the two fixed points with all other cycle contributions cancelled exactly The 1C0 is the spectral determinant for the transfer operator like the one in 1519 with the T00 0 and in general 1 lt1gt 1272 1273 4k A A f WW3 1A121A f2 k 2 Z 1 1 A1644 A2k2 S45 The factor 716 arises because both stabilities A1 and A2 include a factor 7A from the right branch of the map soluCount r 8oct2003 boyscout Versionll 9 2 Aug 21 2007 Cycle expansions Recycle It s the Law Poster New York City Department of Sanitation The Euler product representations of spectral determinants 179 and dynamical zeta functions 1715 are really only a shorthand notation 7 the zeros of the individual factors are not the zeros of the zeta func7 tion and convergence of such objects is far from obvious Now we shall give meaning to the dynamical zeta functions and spectral deter minants by expanding them as cycle expansions series representations ordered by increasing topological cycle length with products in 179 1715 expanded as sums over pseudocycles products of tp s The zeros of correctly truncated cycle expansions yield the desired eigenvalues and the expectation values of observables are given by the cycle aver aging formulas obtained from the partial derivatives of dynamical zeta functions or spectral determinants 181 Pseudocycles and shadowing How are periodic orbit formulas such as 1715 evaluated We start by computing the lengths and stability eigenvalues of the shortest cycles This always requires numerical work such as the Newton s method searches for periodic solutions we shall assume that the numerics is under control and that all short cycles up to a given topological length have been found Examples of the data required for application of perii odic orbit formulas are the lists of cycles given in Tables 203 and 7 It is important not to miss any short cycles as the calculation is as accurate as the shortest cycle dropped 7 including cycles longer than the short est omitted does not improve the accuracy more precisely improves it but painfully slowly Expand the dynamical zeta function 1715 as a formal power series 2 Filmmph 71 1tmtm mt 181 1ltHlt1etgt17 where the prime on the sum indicates that the sum is over all distinct nonirepeating combinations ofprime cycles As we shall frequently use such sums let us denote by t7r 71 17 it 71c an element of the set of all distinct products of the prime cycle weights tp The formal 181 Pseudocycles and shadowing 267 9 N Construction of cycle expansions 269 183 Cycle formulas for dynamical averr 274 184 Cycle expansions for nite alphar bets 277 185 Stability ordering of cycle expanr sions 278 186 Dirichlet series 282 Summary 283 Further reading 284 Exercises 285 References 287 268 CHAPTER 18 CYCLE EXPANSIONS power series 181 is now compactly written as yang1 182 For k gt 1 t7r are weights ofpseudocycles they are sequences of shorter cycles that shadow a cycle with the symbol sequence plpg 1016 along segments 171 p2 pk 2 denotes the restricted sum for which any given prime cycle p contributes at most once to a given pseudocycle weight tw The pseudocycle weight ie the product ofweights 1710 of prime cycles comprising the pseudocycle l t7r 7lk1me Aquot 5Tquot2 quot7 183 depends on the pseudocycle topological length 717 integrated observ7 able AW period T7 and stability A7r n7r 711717771 TWTpTpk A7 ApAp7 AApAmAp 184 Throughout this text the terms periodic orbit and cycle are used interchangeably while periodic orbit is more precise cycle which has many other uses in mathematics is easier on the ear than quotpseudo7 periodic7orbit While in Soviet times acronyms were a rage and in France they remain so we shy away from acronyms such as UPOs Unstable Periodic Orbits 1811 Curvature expansions The simplest example is the pseudocycle sum for a system described by a complete binary symbolic dynamics In this case the Euler product 1715 is given by 14 17t017t117t0117t00117t011 185 1 i 750001 1 i 7500111 i 7501111 7500001 1 7500011 15001011 t001111t010111t011118 see Table 101 and the rst few terms of the expansion 182 ordered by increasing total pseudocycle length are 1lt 1 750 i 751 i 7501 i 75001 i 15011 i 150001 i 150011 i 150111 i 8 u 1950751 1507501 1501751 75075001 75075011 75001751 15011751 71010111 7 186 We refer to such series representation of a dynamical zeta function or a spectral determinant expanded as a sum over pseudocycles and or7 dered by increasing cycle length and instability as a cycle expansion recycle VKOaugZWE ChaosBook mgversmnll a 2 Aug 212007 182 CONSTRUCTION OF CYCLE EXPANSIONS 269 The next step is the key step regroup the terms into the dominant fundamental contributions if and the decreasing curvature corrections 8 each 8 split into prime cycles p of length npn grouped together with pseudocycles whose full itineraries build up the itinerary ofp For the binary case this regrouping is given by 1M 1 750 i 751 i 7501 i 751750 f 75001 f 7501750 75011 7501751 60001 75075001 750111 75011751 t0011 75001751 75075011 t0t01t1l Ht 1 7 E if 7 Z A f n 187 All terms in this expansion up to length np 6 are given in Table 181 We refer to such regrouped series as curvature expansions Such separation into fundamental and curvature parts of cycle expansions is possible only for dynamical systems whose symbolic dy namics has nite grammar The fundamental cycles t0 t1 have no shorter approximants they are the building blocks of the dyname ics in the sense that all longer orbits can be approximately pieced to gether from them The fundamental part of a cycle expansion is given by the sum of the products of all noneintersecting loops of the associ ated Markov graph The terms grouped in brackets are the curvae ture corrections the terms grouped in parenthesis are combinations of longer cycles and corresponding sequences of shadowing pseudocye cles If all orbits are weighted equally tp 2 such combinations cancel exactly and the dynamical zeta function reduces to the topologe ical polynomial 1321 If the ow is continuous and smooth orbits of similar symbolic dynamics will traverse the same neighborhoods and will have similar weights and the weights in such combinations will al most cancel The utility of cycle expansions ofdynamical zeta functions and spectral determinants lies precisely in this organization into nearly cancelin 39 39 c c e 1 39 are dominated by short cyi cles with long cycles giving exponentially decaying corrections In the case where we know of no nite grammar symbolic dyname ics that would help us organize the cycles the best thing to use is a stability cutoffwhich we shall discuss in Section 185 The idea is to truncate the cycle expansion by including only the pseudocycles such that lApl Apkl g Amax with the cutoffAmaX equal to or greater than the most unstable A1 in the data set 182 Construction of cycle expansions 1821 Evaluation of dynamical zeta functions Cycle expansions of dynamical zeta functions are evaluated numerie cally by rst computing the weights tp tp s of all prime cycles p of topological length np g N for given xed 5 and s Denote by subscript the ith prime cycle computed ordered by the topological ChaasBook mgversmnll a ZAug212007 recycle 7 soaugzoos 2 Section 13 3 j Seotion18 4 270 CHAPTER 18 CYCLE EXPANSIONS to t1 7 he me i 75100 7510750 75101 7510751 751000 75100750 t1001 75100751 75101750 7517510750 751011 75101751 7510000 751000750 7510001 751001750 751000751 75075100751 7510010 751007510 7510101 751017510 7510011 751011750 751001751 75075101751 7510111 751011751 75100000 7510000750 75100001 7510001750 7510000751 750751000751 75100010 7510010750 7510007510 750751007510 75100011 7510011750 7510001751 750751001751 75100101 75100110 7510010751 7510110750 7510751001 7510075101 750751075101 751751075100 75101110 7510110751 7510117510 751751017510 75100111 7510011751 7510111750 750751011751 75101111 7510111751 Table 181 The binary curvature expansion 187 up to length 6 listed in such way that the sum of terms along the pth horizontal line is the curvature 6 associated with a prime cycle p or a combination of prime cycles such as the t100101 t100110 pair recycle VKOaugZOOE ChaosBook mgvexsmnll a 2 Aug 212007 182 CONSTRUCTION OF CYCLE EXPANSIONS 271 length n g 710 The dynamical zeta function 14 N truncated to the 711 g N cycles is computed recursively by multiplying 1lti 1lti711 7 t02 188 and truncating the expansion at each step to a nite polynomial in 2 n g N The result is the Nth order polynomial approximation N 1ltN 717 Zena 189 711 In other words a cycle expansion is a Taylor expansion in the dummy variable 2 raised to the topological cycle length If both the number of cycles and their individual weights grow not faster than exponentially with the cycle length and we multiply the weight of each cycle p by a factor 27 the cycle expansion converges for suf ciently small If the dynamics is given by iterated mapping the leading zero of 189 as function of 2 yields the leading eigenvalue of the appropriate evolution operator For continuous time ows 2 is a dummy variable that we set to z 1 and the leading eigenvalue of the evolution opera tor is given by the leading zero of 189 as function of 5 1822 Evaluation of traces spectral determinants Due to the lack offactorization of the full pseudocycle weight dEt 1 7 1 1171172Tquot dEt 1 7 MP1 dEt 1 7 MP27 the cycle expansions for the spectral determinant 179 are somewhat less transparent than is the case for the dynamical zeta functions We commence the cycle expansion evaluation of a spectral determine ant by computing recursively the trace formula 1610 truncated to all prime cycles p and their repeats such that npr g N nrgN tr zL tr zL Z 6539Alt2gt5Tlt2T my 7 7 718 2 1 2L N trliz N gong 071rc 1810 This is done numerically the periodic orbit data set consists of the list of the cycle periods Tp the cycle stability eigenvalues A117 A173 7 AW and the cycle averages of the observable A17 for all prime cycles p such that 711 g N The coef cient of 2 is then evaluated numerically for the given 5 s parameter values Now that we have an expansion for the trace formula 169 as a power series we compute the Nth order approximation to the spectral determinant 173 N det17 2 N 17 Z ann Q 7 nth cumulant 1811 711 ChaasBook mgvexsmnll a ZAug212007 recycle 7 soaugzoos W Chapter 739 272 CHAPTER 18 CYCLE EXPANSIONS as follows The logarithmic derivative relation 174 yields tr gt det17 zL 72det17 2L Q12 2Q222 3Q32339 39 21 172 0120222quot391 QIZ QZZQ quot39 so the nth order term ofthe spectral determinant cycle or in this case the cumulant expansion is given recursively by the trace formula ex7 pansion coef cients Q g on 7 ole 701Qn717 Q1 7 01 1812 Given the trace formula 1810 truncated to 2N we now also have the spectral determinant truncated to z The same program can also be reused to compute the dynamical zeta in 1810 by the product of expanding eigenvalues Am He Aie see Sec7 tion 173 The calculation of the leading eigenvalue ofa given continuous ow evolution operator is now straightforward After the prime cycles and the pseudocycles have been grouped into subsets of equal topological length the dummy variable can be set equal to z 1 With 2 1 expansion 1811 is the cycle expansion for 176 the spectral deter7 minant det s 7 A We vary 3 in cycle weights and determine the eigenvalue 50 by nding 3 50 for which 1811 vanishes As an example the convergence of a leading eigenvalue for a nice hyperbolic system is illustrated in Table 182 by the listing of pinball escape rate 7 estimates computed from truncations of 187 and 1811 to different maximal cycle lengths The pleasant surprise is that the coef cients in these cycle expansions can be proven to fall off exponentially or even faster due to analyticity of det s 7 A or 1 s for 3 values well beyond those for which the corresponding trace formula diverges function cycle expansion 189 by replacing H lt1 7 AZ j 1823 Newton algorithm for determination of the evolution operator eigenvalues lt9 The cycle expansions of spectral determinants yield the eigen7 values of the evolution operator beyond the leading one A convenient way to search for these is by plotting either the absolute magnitude ln ldet s 7 A or the phase of spectral determinants and dynamical zeta functions as functions of the complex variable 5 The eye is guided to the zeros of spectral determinants and dynamical zeta functions by means of complex 3 plane contour plots with different intervals of the absolute value of the function under investigation assigned different colors zeros emerge as centers of elliptic neighborhoods of rapidly changing colors Detailed scans of the whole area of the complex 3 recycle VKOaugZWE ChaosBook mgversionll a 2 Aug 212007 182 CONSTRUCTION OF CYCLE EXPANSIONS 273 Rla N 1 detsiA 1 lCs 1 1 s37disk 1 1 039 0407 2 04105 041028 0435 3 0410338 0410336 04049 6 4 04103384074 04103383 040945 5 04103384077696 04103384 0410367 6 0410338407769346482 04103383 0410338 7 04103384077693464892 04103396 8 0410338407769346489338468 9 04103384077693464893384613074 10 04103384077693464893384613078192 041 072 0675 067797 0677921 06779227 06779226894 06779226896002 0677922689599532 067792268959953606 oomwmmugtwm Table 182 37disk repellen J r t d from t p 39 rfthe spectral determinant 176 and the dynamical zeta function 1715 as function of the maximal cycle length N The rst column indicates the diskidisk center separation to disk radius ratio Rza the second column gives the maximal cycle length used and the third the estimate ofthe classical escape rate from the fun damental domain spectral determinant cycle expansion As for larger diskidisk r 39 39 i muneuniform g i uetter for Rza 6 than for Rza 3 For comparison the fourth column lists a few estimates from from the fundamental domain dynamical zeta function cycle expansion 187 and the fth from the full 37disk cycle expansion 1836 The convergence of the fundamental domain dynamical zeta function is signi cantly slower than the convergence of the corresponding spectral determinant and the full uni factorized 37disk dynamical zeta function has still poorer convergence RE Rosenqvist ChaasBook mgvexsmnll a ZAug212007 recycle 7 soaugzoos Fig 181 Examples ofthecomplex s plane sca contour plots of the logarithm of the absolute values ofa 1 b spec tral determinant dets 7 A for the 37 disk system separation o R 6 A1 subspace are evaluated numerically The eigenvalues of the evolution operator 1 are given by the centers of elliptic neighr borhoods ofthe rapi y narrowing rings families ofzeros RE Rosenqvist B FBsf0 line Fig 182 The eigenvalue condition is Sat is ed onthecurveF 0the 3 s The expectation va u 1512 is given by the slope ofthe curve 274 CHAPTER 18 CYCLE EXPANSIONS I H l AIMl 39 39 E a guma AM u H W M plane under investigation and searches for the zeros of spectral deters minants Fig 181 reveal complicated patterns of resonances even for something so simple as the Ssdisk game of pinball With agood starting guess such as alocation of a zero suggested by the complex 5 scan of Fig 181 a zero 1s 0 can now be easily determined by standard numerical methods such as the iterative Newton algorithm 124 with the mth Newton estimate given by 71 5ml 5m 7 ltltSm ltilsmgt 5711 1813 M T The dominator required for the Newton iteration is given below by the cycle expansion 1822 We need to evaluate it anyhow as T4 enters our cycle averaging formulas 183 Cycle formulas for dynamical averages The eigenvalue condition in any of the three forms that we have given so far 7 the level sum 7 the dynamical zeta function 182 the spec tral determinant 1811 H 1 Zn ttl s mm 1814 o 172 t t z ss 1815 0 1762 QnQn ss s 1816 m1 is an implicit equation for the eigenvalue 5 5 ofform F 7 s 0 The eigenvalue 5 5 as afunction of 13 is sketched in Fig 182 the eigenvalue condition is satis ed on the curve F 0 The cycle ave eraging formulas for the slope and the curvature of 5 are obtained as recycle smugzuus ChmsBmkorgvemonH 92Aug 212cm 183 CYCLE FORMULAS FOR DYNAMICAL AVERAGES 275 in 1512 by taking derivatives of the eigenvalue condition Evaluated along F 0 the rst derivative leads to d 0 FWSW 6F d3 6F d3 6F 6F gs E m and the second derivative of F 7 55 0 yields dQS 82F d3 82F d3 262F 8F T52 752 t 3533 g E 1818 Denoting by 6F 6F ltAgtF i 7 7 i 7 95 mam 93 mam 62F ltA7 ltAgt2gtF 672 1819 ss respectively the mean cycle expectation value of A the mean cycle pe riod and the second derivative of F computed for F 7 55 0 we obtain the cycle averaging formulas for the expectation value of the ob servable 1512 and its variance a amp lt gt F 1820 ltlta7ltagtgtgt mltltA7ltAgtgtgtF 1821 These formulas are the central result of the periodic orbit theory As we shall now show for each choice ofthe eigenvalue condition function F 7 s in 7 182 and 1811 the above quantities have explicit cycle expansions 1831 Dynamical zeta function cycle expansions For the dynamical zeta function condition 1815 the cycle averaging formulas 1817 1821 require evaluation of the derivatives of dyni amical zeta function at a given eigenvalue Substituting the cycle ex pansion 182 for dynamical zeta function we obtain A4 7 214th 1822 a l E 1 T I Z thw n I izgg Z nwtw where the subscript in 4 stands for the dynamical zeta function av erage over prime cycles AW T7 and n7r are evaluated on pseudocycles ChaasBook mgversionll a ZAug212007 recycle 7 soaugzoos 276 CHAPTER 18 CYCLE EXPANSIONS 184 and pseudocycle weights t7r t7r 257 55 are evaluated at the eigenvalue 55 In most applications 5 0 and 55 of interest is typically the leading eigenvalue 50 50 0 of the evolution generator Eor bounded ows the leading eigenvalue the escape rate vanishes 50 0 the exponent 5A7r 7 5T7r in 183 vanishes so the cycle expani sions take a simple form 7 7 k1AP1Ap2quot39APk ltAgtCAZlt 1 Am Am 1823 and similarly for T For example for the complete binary symi bolic dynamics the mean cycle period T4 is given by 7r T0 T1 lt T01 T0T1gt 7 7 1824 14 lel w Moll leAll l lt T001 7 T01 T0gt lt T011 7 T01 T1 le011 1A01Aol lenl lelAll H Note that the cycle expansions for averages are grouped into the same shadowing combinations as the dynamical zeta function cycle expan7 sion 187 with nearby pseudocycles nearly cancelling each other The cycle averaging formulas for the expectation value of the observe able a follow by substitution into 1821 Assuming zero mean drift a 0 the cycle expansion 1811 for the variance A 7 14124 is given by 2 ltA2gt4 Z 71k1Ap1 AP392I39I39 39 Apk 1825 Wu Apkl 1832 Spectral determinant cycle expansions The dynamical zeta function cycle expansions have a particularly sim7 ple structure with the shadowing apparent already by a termibyiterm inspection of Table 182 For nice hyperbolic systems the shadowing ensures exponential convergence of the dynamical zeta function cycle expansions This however is not the best achievable convergence As has been explained in Chapter 7 for such systems the spectral deti erminant constructed from the same cycle data base is entire and its cycle expansion converges faster than exponentially In practice the best convergence is attained by the spectral determinant cycle expan7 sion 1816 and its derivatives The 885 685 derivatives are in this case computed recursively by taking derivatives of the spectral deter minant cycle expansion contributions 1812 and 1810 The cycle averaging formulas are exact and highly convergent for nice hyperbolic dynamical systems An example of its utility is the cyi cle expansion formula for the Lyapunov exponent of Example 181 Eure ther applications of cycle expansions will be discussed in Chapter 7 recycle VKOaugZOOE ChaosBook mgversmnll a 2 Aug 212007 184 CYCLE ECPANSIONS FOR FlNlTE ALPHABETS Z77 1833 Continuous vs discrete mean return time Sometimes it is convenient to compute an expectation value along a vary Wildly and it is not at all clear that the continuous and discrete time averages are related in any simple Way The relationship turns on to be both elegantly simple and totally general The mean cycle period T4 xes the normalization of the unit of time it can be interpreted as the average near recurrence or the average rst return time For example if We have evaluated a billiard expectar tion value a in terms of continuous time and would like to also have the corresponding average ltagtdm measured in discrete time given by the number of reflections off billiard Walls the two averages are related by ltagtdscr W e We gt 1826 Where lt12 is the average of the number of bounces up along the cycle Example 181 Cycle expansion formula for Lyapunov exponents n Section 153 we de ned the Lyapunov exponent for a 1d mapping re lated it to the leading eigenvalue of an evolution operator and promised to evaluate it Now we are nally in positionto deliver on our promise m me Lyapunov exponent in terms of prime cycles A iciloglAeil10gleol 1827 a quot39Apkl lt71 m For a repeller the 1lApl weights are replaced by normalized measure 2 expynp1Apl where 39y isthe escape rate We mention here Without proof that for 2d Hamiltonian flows such as our game of pinball there is only one expanding eigenvalue and 1827 applies as it stands m depth chaptenz p 77 184 Cycle expansions for nite alphabets a A nite Markov graph like the one given in Fig 1331 d is a compact encoding of the transition or the Markov matrix for a given subshift It is a sparse matrix and the associated determinant 1317 can be Written down by inspection it is the sum of all possible pare titions of the graph into products of nonrintersecting loops with each loop carrying a minus sign 1610 T 1 to ifoon tooot t00011 tot0011 t0011f0001 1828 We GhosBaoknrgvasml 19 2 an 2i m 278 CHAPTER 18 CYCLE EXPANSIONS The simplest application of this determinant is to the evaluation of the topological entropy if we set tp 271 where np is the length of the p7cycle the determinant reduces to the topological polynomial 1318 The determinant 1828 is exact for the nite graph Fig 1331 e as well as for the associated nite7dimensional transfer operator of Exam7 ple 151 For the associated in nite dimensional evolution operator it is the beginning of the cycle expansion of the corresponding dynamical zeta function 1 1 i 750 i 150011 i 150001 750001750011 7t00011 7t0t0011 curvatures 1829 The cycles 5 m and m are the fundamental cycles introduced in 187 they are not shadowed by any combinations of shorter cycles and are the basic building blocks of the dynamics All other cycles appear together with their shadows for example the town 7 totoou 39 and yield 1 39 small corrections for hyperbolic systems For the cycle counting purposes both tab and the pseudocycle com7 bination tab tatb in 182 have the same weight zndnb so all cur7 vature combinations tab 7 tatb vanish exactly and the topological poly7 nomial 1321 offers a quick way ofchecking the fundamental part ofa cycle expansion Since for nite grammars the topological zeta functions reduce to polynomials we are assured that there are just a few fundamental cy7 cles and that all long cycles can be grouped into curvature combina7 tions For example the fundamental cycles in Exercise 92 are the three 27cycles which bounce back and forth between two disks and the two 37cycles which visit every disk It is only after these fundamental cycles have been included that a cycle expansion is expected to start converg7 ing smoothly ie only for n larger than the lengths of the fundamen7 tal cycles are the curvatures 8 in expansion 187 a measure of the deviations between long orbits and their short cycle approximants ex7 pected to fall off rapidly with n 185 Stability ordering of cycle expansions There is never a second chance Most often there is not even the rst chance John Wilkins CP Dettmann and P Cvitanovic Most dynamical systems of interest have no nite grammar so at any order in z a cycle expansion may contain unmatched terms which do not t neatly into the almost cancelling curvature corrections Similarly for intermittent systems that we shall discuss in Chapter 7 curvature corrections are in general not small so again the cycle expansions may recycle VKOaugZWE ChaosBook mgvexsionll a 2 Aug 212007 185 STABILITY ORDERING OF CYCLE EXPANSIONS 279 converge slowly For such systems schemes which collect the pseudo cycle terms according to some criterion other than the topology of the ow may converge more quickly than expansions based on the topoi logical length All chaotic systems exhibit some degree of shadowing and a good truncation criterion should do its best to respect the shadowing at least approximately If a long cycle is shadowed by two or more shorter cycles and the ow is smooth the period and the action will be addii tive in sense that the period of the longer cycle is approximately the sum of the shorter cycle periods Similarly stability is multiplicative so shadowing is approximately preserved by including all terms with pseudocycle stability lAP1quot39Apkl S Amax 0830 and ignoring all more unstable pseudocycles Two such schemes for ordering cycle expansions which approximately respect shadowing are truncations by the pseudocycle period or ac tion and the stability ordering that we shall discuss here In these schemes a dynamical zeta function or a spectral determinant is expanded keeping all terms for which the period action or stability for a combi7 nation of cycles pseudocycle is less than a given cutoff The two settings in which the stability ordering may be preferable to the ordering by topological cycle length are the cases of bad grammar and of intermittency 1851 Stability ordering for bad grammars For generic ows it is often not clear what partition of the state space generates the optimal symbolic dynamics Stability ordering does not require understanding dynamics in such detail if you can nd the cycles you can use stability ordered cycle expansions Stability truni cation is thus easier to implement for a generic dynamical system than the curvature expansions 187 which rely on nite subshift approxii mations to a given ow Cycles can be detected numerically by searching a long trajectory for near recurrences The long trajectory method for detecting cycles pref erentially nds the least unstable cycles regardless of their topological length Another practical advantage of the method in contrast to New ton method searches is that it only nds cycles in a given connected ergodic component of state space ignoring isolated cycles or other ere godic regions elsewhere in the state space Why should stability ordered cycle expansion of a dynamical zeta function converge better than the rude trace formula 7 The argu ment has essentially already been laid out in Section 137 in truncations that respect shadowing most of the pseudocycles appear in shadowing combinations and nearly cancel while only the relatively small subset affected by the longer and longer pruning rules is not shadowed So the error is typically of the order of 1A smaller by factor ehT than ChaasBook mgversmnll a ZAug212007 recycle 7 soaugzoos 280 CHAPTER 18 CYCLE EXPANSIONS the trace formula 7 error where h is the entropy and T typical cycle length for cycles of stability A 1852 Smoothing 6 The breaking of exact shadowing cancellations deserves further comment Partial shadowing which may be present can be partially restored by smoothing the stability ordered cycle expansions by re placing the lA weight for each term with pseudocycle stability A Api Apk by fAA Here fA is a monotonically decreasing func7 tion from f0 l to fAmaX 0 No smoothing corresponds to a step function A typical shadowing error induced by the cutoff is due to two pseudocycles of stability A separated by AA and whose contribution is of opposite signs lgnoring possible weighting factors the magnitude of the resulting term is of order lAi lA AA m AAA2 With smooth ing there is an extra term of the form f AAAA which we want to minimise A reasonable guess might be to keep f AA constant and as small as possible that is fA 1 7 The results of a stability ordered expansion 1830 should always be tested for robustness by varying the cutoff Amax If this introduces signi cant variations smoothing is probably necessary 1853 Stability ordering for intermittent ows e Longer but less unstable cycles can give larger contributions to a cycle expansion than short but highly unstable cycles In such situ7 ation truncation by length may require an exponentially large number ofvery unstable cycles before a signi cant longer cycle is rst included in the expansion This situation is best illustrated by intermittent maps that we shall study in detail in Chapter 7 the simplest of which is the Farey map f0z 171 0S1S12 161 l f11Ezgta Was1 1831 a map which will reappear in the intermittency Chapter 7 and in Chapter 7 in context of circle maps For this map the symbolic dynamics is of complete binary type so lack of shadowing is not due to lack of a nite grammar but rather to the intermittency caused by the existence of the marginal xed point x0 0 for which the stability equals A0 1 This xed point does not participate directly in the dynamics and is omitted from cycle expani sions Its presence is felt in the stabilities of neighboring cycles with recycle VKOaugZOOE ChaosBook mgversionll a 2 Aug 212007 185 STABILITY ORDERING OF CYCLE EXPANSIONS 281 10 100 1000 10000 N A mm Fig 183 Comparison of cycle expansion truncation schemes for the Farey map 1831 the deviation of the truncated cycles expansion for 11N01 from the exact ow con servation value 10 0 is a measure of the accuracy of the truncation The jagged line is logarithm of the stability ordering truncation error the smooth line is smoothed according to Section 1852 the diamonds indicate the error due the topological length truncation with the maximal cycle length N shown They are placed along the stability cutoff axis at points determined by the condition that the total number of cycles is the same for both truncation schemes n consecutive repeats of the symbol 05 whose stability falls of only as A N n2 in contrast to the most unstable cycles with n consecutive 1 s which are exponentially unstable A014 N 122 The symbolic dynamics is of complete binary type A quick count in the style of Section 1352 leads to a total of 74248450 prime cycles of length 30 or less not including the marginal point 10 0 Evalui ating a cycle expansion to this order would be no mean computational feat However the least unstable cycle omitted has stability of roughly A1020 N 302 900 and so amounts to a 01 correction The situation may be much worse than this estimate suggests because the next 1031 cycle contributes a similar amount and could easily reinforce the error Adding up all such omitted terms we arrive at an estimated error of about 3 for a cycleilength truncated cycle expansion based on more than 10g pseudocycle terms On the other hand truncating by stability at say Amax 3000 only 409 prime cycles suf ce to attain the same accuracy of about 3 error Fig 1853 As the Farey map maps the unit interval onto itself the leading eigen7 value ofthe PerroniFrobenius operator should equal so 0 so 1 0 0 Deviation from this exact result serves as an indication of the con vergence of a given cycle expansion The errors of different truncation schemes are indicated in Fig 1853 We see that topological length truni cation schemes are hopelessly bad in this case stability length truncai tions are somewhat better but still rather bad In simple cases like this one where intermittency is caused by a single marginal xed point the convergence can be improved by going to in nite alphabets ChaasBook mgversionll a ZAug212007 recycle 7 soaugzoos 282 CHAPTER 18 CYCLE EXPANSIONS 186 Dirichlet series The most patient reader will thank me for compressing so much nonsense and falsehood into a few lines Gibbon 6 A Dirichlet series is de ned as 103 Zajews 1832 j1 where s aj are complex numbers and M is a monotonically increas ing series of real numbers A1 lt A2 lt lt 1 lt A classical exam ple of a Dirichlet series is the Riemann zeta function for which aj l M lnj In the present context formal series over individual pseudo cycles such as 182 ordered by the increasing pseudocycle periods are often Dirichlet series For example for the pseudocycle weight 183 the Dirichlet series is obtained by ordering pseudocycles by increasing periods A7 Tpl TI2 Tpk with the coef cients e 39AP1AP239 APk d 04W 7r AplAPZ 7 where d7r is a degeneracy factor in the case that d7r pseudocycles have the same weight If the series 2 a1 diverges the Dirichlet series is absolutely convere gent for 995 gt aa and conditionally convergent for 995 gt ac where aa is the abscissa of absolute convergence N 1 0a ngnm sup E1n 11 1833 and 00 is the abscissa of conditional convergence N 1 a ngnoosupmln F2111 1834 We shall encounter another example of a Dirichlet series in the semi classical quantization Chapter 7 where the inverse Planck constant is a complex variable 3 ih A7 SP1 SP2 Sp is the pseudo cycle action and a7r l AmX A times possible degeneracy and topological phase factors As the action is in general not a line ear function of energy except for billiards and for scaling potentials where a variable 3 can be extracted from Sp semiclassical cycle expane sions are Dirichlet series in variable 3 ih but not in E the complex energy variable recycle VKOaugZOOE ChaosBook mgversmnll a 2 Aug 212007 186 DIRICHLET SERIES 283 Summary A cycle expansion is a series representation of a dynamical zeta func7 tion trace formula or a spectral determinant with products in 1715 7 expanded as sums over pseudocycles products of the prime cycle weights tp If a ow is hyperbolic and has a topology of a Smale horseshoe a subshift of nite type the dynamical zeta functions are holomorphic the spectral determinants are entire and the spectrum of the evolu7 tion operator is discrete The situation is considerably more reassuring than what practitioners of quantum chaos fear there is no abscissa of absolute convergence and no entropy barier the exponential pro liferation of cycles is no problem spectral determinants are entire and converge everywhere and the topology dictates the choice of cycles to be used in cycle expansion truncations In that case the basic observation is that the motion in dynamical systems of few degrees of freedom is in this case organized around a few fundamental cycles with the cycle expansion of the Euler product 1lt17tf7am f n regrouped into dominant fundamental contributions if and decreasing curvature corrections en The fundamental cycles tf have no shorter approximants they are the building blocks of the dynamics in the sense that all longer orbits can be approximately pieced together from them A typical curvature contribution to en is a difference of a long cycle ab minus its shadowing approximation by shorter cycles a and 12 tab tat tab1 tatbtab The orbits that follow the same symbolic dynamics such as ab and a pseudocycle ab lie close to each other have similar weights and for longer and longer orbits the curvature corrections fall off rapidly Indeed for systems that satisfy the axiom A requirements such as the Sidisk billiard curvature expansions converge very we Once a set of the shortest cycles has been found and the cycle pee riods stabilities and integrated observable computed the cycle aver aging formulas such as the ones associated with the dynamical zeta function ltagt ltAgt4 ltTgt4 a l E 1 ltAgt4 g Z Awtm ltTgt4 EE Z T77r yield the expectation value the chaotic ergodic average over the non wandering set of the observable az ChaasBook mgversmnll a ZAug212007 recycle 7 soaugzoos 284 Further reading Further reading Pseudocycle expansions Bowen s introduction of shade owing eipseudoorbits 23 was a signi cant contribution to Smale s theory Expression pseudoorbits seems to have been introduced in the Parry and Pollicott s 1983 pa per 5 Following them M Berry 9 had used the expres7 sion pseudoorbits in his 1986 paper on Riemann zeta and quantum chaos Cycle and curvature expansions of dynamical zeta functions and spectral determinants were introduced in Refs 10 2 Some literature 14 refers to the pseudoorbits as composite orbits and to the cycle expansions as Dirichlet series see also Remark 186 and Section 186 Cumulant expansion To a statistical mechanician the curvature expansions are very reminiscent of cumui lant expansions Indeed 1812 is the standard Plemelji Smithies cumulant formula 7 for the Fredholm determii nant discussed in more detail in Appendix 7 The dif ference is that in cycle expansions each QT coef cient is expressed as a sum over exponentially many cycles Exponential growth of the number of cycles Go ing from NT z Nquot periodic points of length n to MT prime cycles reduces the number of computations from NT to MT z Nquot 1n Use of discrete symmetries Chap ter 7 reduces the number of nth level terms by another factor While the reformulation of the theory from the trace 1628 to the cycle expansion 187 thus does not eliminate the exponential growth in the number of cycles in practice only the shortest cycles are used and for them the computational labor saving can be signi cant Shadowing cycleibyicycle A glance at the low order curvatures in the Table 181 leads to the temptai tion of associating curvatures to individual cycles such as 80001 730001 7 totem Such combinations tend to be nu recycle VKOaugZOOE merically small see for example Ref 3 table 1 How ever splitting 6 into individual cycle curvatures is not possible in general 20 the rst example of such ambigui ity in the binary cycle expansion is given by the tioowi tloono O lt gt 1 symmetric pair of 67cycles the counterterm tomton in Table 181 is shared by the two cycles Stability ordering The stability ordering was intro duced by Dahlqvist and Russberg 12 in a study of chaotic dynamics for the 21121 a potential The presentation here runs along the lines of Dettmann and Morriss 13 for the Lorentz gas which is hyperbolic but the symi bolic dynamics is highly pruned and Dettmann and Cvii tanovic 14 for a family of intermittent maps In the appli7 cations discussed in the above papers the stability order ing yields a considerable improvement over the topologii cal length ordering In quantum chaos applications cycle expansion cancelations are affected by the phases of pseui docycles their actions hence period ordering rather than stability is frequently employed Are cycle expansions Dirichlet series Even though some literature 14 refers to cycle expani sions as Dirichlet series they are not Dirichlet series Cycle expansions collect contributions of individual cycles into groups that correspond to the coef cients in cumui lant expansions of spectral determinants and the conver7 gence of cycle expansions is controlled by general prop erties of spectral determinants Dirichlet series order cyi cles by their periods or actions and are only conditionally convergent in regions of interest The abscissa of absolute convergence is in this context called the entropy barrier 1 contrary to the frequently voiced anxieties this number does not necessarily has much to do with the actual con vergence of the theory ChaosBook mgversmnll a 2 Aug 212007 Exercises 285 1 Exercises 181 Cycle expansions Write programs that implei b Show that 182 188 ment binary symbolic dynamics cycle expansions for a dynamical zeta functions b spectral deter minants Combined with the cycles computed for a 27branch repeller or a 37disk system they will be useful in problem that follow Escape rate for a lid repeller Continuation of Exercise 171 7 easy but lon Consider again the quadratic map 1731 Azl 7 x on the unit interval for de nitiveness take A 6 Describing the itinerary of any trajectory by the bi nary alphabet 0 l 0 if the iterate is in the rst half of the interval and 1 if is in the second half we have a repeller with a complete binary symbolic dynamics a Sketch the graph of f and determine its two xed points 6 and T together with their stabili ities Sketch the two branches of f l Determine all the prime cycles up to topological length 4 using your pocket calculator and backwards iteration of f see Section 1221 5 Determine the leading zero of the zeta func7 tion 1715 using the weigths tp znPiApi where AF is the stability of the p cycle Show that for A 92 the escape rate of the repeller is 0361509 using the spectral determinant with the same cycle weight If you have taken A 6 the escape rate is in 083149298 as shown in Solution 182 Compare the coef cients of the spectral deti erminant and the zeta function cycle expani sions Which expansion converges faster Per Rosenqvist Escape rate for the Ulam map medium We will try to compute the escape rate for the Ulam map 1222 1 4901 I using the method of cycle expansions The answer should be zero as nothing escapes a Compute a few of the stabilities for this map Show that AD 1 i 4 A001 78 and A011 8 ChaasBook mgversmnll a ZAug212007 184 185 186 187 42quot A61 6 and determine a rule for the sign 5 hard Compute the dynamical zeta function for this system 71litoit1t017tot1i 39 You might note that the convergence as func7 tion of the truncation cycle length is slow Try to x that by treating the A0 4 cycle sepa7 rately Pinball escape rate semiianalytical Estimate the 37disk pinball escape rate for R a 6 by substii tuting analytical cycle stabilities and periods Exeri cise 93 and Exercise 94 into the appropriate binary cycle expansion Compare with the numerical esti7 mate Exercise 153 Pinball escape rate from numerical cycles Com pute the escape rate for R a 6 37disk pinball by substituting list of numerically computed cycle stai bilities of Exercise 125 into the binary cycle expani SlOn Pinball resonances in the complex plane Plot the logarithm of the absolute value of the dynami ical zeta function and or the spectral determinant cycle expansion 185 as contour plots in the com plex 5 plane Do you nd zeros other than the one corresponding to the complex one Do you see ev7 idence for a nite radius of convergence for either cycle expansion Counting the Sidisk pinball counterterms Vere ify that the number of terms in the 37disk pinball curvature expansion 1835 is given by Hlt1tp p 17351 7 M m ig g z 1 3 2 6 125 206 This means that for example as has a total of 20 te ms in agreement with the explicit 37disk cycle expansion 1836 exaReqc r ianovzoov 46 12 2 173z2 7 2Z3 487 848 184 188 189 286 37disk unfactorized zeta cycle expansions Check that the curvature expansion 182 for the 37disk pinball assuming no symmetries between disks is given by 1g l 7 z2t12l 7 z2t13l 7 z2t23 17 Zat12317 Zat1321 7 Z4t1213 17 Z4t123217 Z4t132317 25t12123 39 quot 17 Z2t12 7 Z2t23 7 Z2t317 23t123 t132 1810 Exercises where in the coef cient to Z6 the abbreviations 33 and S4 stand for the sums over the weights of the 12 orbits with multiplicity 8 and the 5 orbits of mul7 tiplicity 4 respectively the orbits are listed in Ta7 ble 134 Tail resummations A simple illustration of such tail resummation is the C function for the Ulam map 1222 for which the cycle structure is exceptionally simple the eigenvalue of the x0 0 xed point is 7511031213 7 t12t13 1232 7 t12t23 751323 7 tl ktwlilile the eigenvalue of any other n7cycle is i2quot 7Z5t12123 7 t12t123 quot 7 39 quot The symmetrically arranged 37disk pinball cycle ex7 pansion of the Euler product 182 see Table 134 and Fig 93 is given by 1g l 7 22t123l 7 z3t123217 24t12133 17 z5t12123617 Z6t121213617 Z6t12132331 Tle cycle weights used in thermodynamic av7 eraging are to 4Tz t1 t 22 t tquot for p 7 0 The simplicity of the cycle eigenvalues en7 ables us to evaluate the C function by a simple trick we note that if the value of any n7cycle eigenvalue were tquot 1721 would yield 1g 1 7 2t There is only one cycle the x0 xed point that has a dif7 n weight 1 7 to so we factor it out multiply 2 3 4 2 5 fere t 7 17 3 112 7 2 ma 7 3z mm 7 t127 6z tlm ew wl 7 t17 t and obtain a rational g Zs 6 75121213 375121323 2 7 9t12t1213 7 23 function 7627 t1212123 t1212313 t1213123 tfztma 7 3t12t12123 7 t123t1213 7328 2t12121213 t12121313 2t12121323 2t12123123 1C z 2 t12123213 t12132123 1 3t12t1213 t12t123 175105107123 18139 2 6t12t121213 7 3t12t121323 7 4t123t12123 7 t12 b sid r ho7178v would have detected the pole at Unsymmetrized cycle expansions The above 37disk cycle expansions might be useful for cross7 checking purposes but as we shall see in Chap7 ter 7 they are not recommended for actual compu7 tations as the factorized zeta functions yield much better convergence 47disk unfactorized dynamical zeta function cycle expansions For the symmetriclly arranged 47disk pinball the symmetry group is C40 of order 8 The degenerate cycles can have multiplicities 2 4 or 8 see Table 132 1g 72472273 874 garquot l zt12 1 zt13 l zt123 1 zt1213l gym amp39 z 11 without the above trick As the 6 xed point is isolated in its stability we would have kept the factor 1 7 to in 187 unexpanded and noted that all curvature combinations in 187 which include the to factor are unbalanced so that the cycle ex7 pansion is an in nite series H17tp 171017t7t27t37t47 F 1840 we shall return to such in nite series in Chap7 7 The glelometric series in the brackets sums ad we expanded the l 7 to factor 1 7 24t1234217 z4t124341 7 z5t12123817 ZSM ul h wmgt i that the ratio of the succes7 17 25t12143817 25t12313817 Z5t12lt113gt8 39 quot and the cycle expansion is given by 1g 17 z24t12 2t137 8z3t123 1811 Escape rate for the Rossler system sive curvature ligg ctly an an t summing we would recover the rational C function 1839 continuation of Exercise 127 Try to compute the escape rate for 7Z48t1213 4731214 2 731234 4731243 7 67312 7 tfdle dsslegkystem 215 using the method of cycle 7825t12123 t12124 t12134 717 t12143 t12313 t1 HmsiP t1EEW ng2 I IOUJd be Zem39 as nmhmg ldteall you should already have computed 74Z62 S8 S4 2 3amp2 t13 t12tfa 7 8t12t1eTEa f12 121 les an have an approximate grammar but 72 t12t1234 7 4t12t1243 7 47313731213 7 2t13t1214 7 faili g lthat you can cheat a bit and peak into Ta7 72 t13t1243 7 723 i 39 quot recycle VKOaugZWE ble 7 1838 ChaosBook mgvexsmnll a 2 Aug 212007 Fixed points and how to get them F Christiansen Having set up the dynamical context now we turn to the key and unavoidable piece of numerics in this subject search for the solutions IT r 6 Rd T6 D2 of the periodic orbit condition f 3 for a given flow or mapping We know from Chapter 7 that cycles are the necessary ingredient for evaluation of spectra of evolution operators In Chapter 10 we have do olnnori quot 39 39 c I I irl m topolog TgtU 02D icall This chapter is intended as a handsron guide to extraction of periodic orbits and should be skipped on rst reading 7 you can return to it whenever the need for nding actual cycles arises understood about the topology of a flow such as in highrdimensional periodic orbit searches fasttrack Chapter 77 p 77 A prime cycle p of period Tp is a single traversal of the periodic or bit so our task will be to nd a cycle pointr 6 p and the shortest time crosses a Poincare section 71p times is a xed point of the P iterate of the Poincare section return map P hence we shall refer to all cycles as values and the period of the cycle are independent of the choice of the initial point so it will suf ce to solve 121 at a single cycle point the cycle is unstable simple integration forward in time will not reveal it and methods to be described here need to be deployed In essence any method for nding a cycle is based on devising a new dynamir cal system which possesses the same cycle but for which this cycle is 12 121 Where arethe cycles7 122 Onedimensionai mappings 123 Multipoint shooting method 124 dedimensional mappings 125 Flows Summary Further reading Exercises References IE chapter 77 3 Section 52 W Chapter 7 158 CHAPTER 12 FIXED POINTS AND HOW TO GET THEM attractive Beyond that there is a great freedom in constructing such systems and many different methods are used in practice Due to the exponential divergence of nearby trajectories in chaotic dynamical systems xed point searches based on direct solution of the xedipoint condition 121 as an initial value problem can be numerii cally very unstable Methods that start with initial guesses for a number of points along the cycle such as the multipoint shooting method de scribed here in Section 123 and the variational methods of Chapter 7 are considerably more robust and safer A prerequisite for any exhaustive cycle search is a good understand ing of the topology of the ow a preliminary step to any serious pee riodic orbit calculation is preparation of a list of all distinct admissible prime periodic symbol sequences such as the list given in Table 101 The relations between the temporal symbol sequences and the spatial layout of the topologically distinct regions of the state space discussed in Chapters 10 and 11 should enable us to guess location of a series of periodic points along a cycle Armed with such informed guess we proceed to improve it by methods such as the NewtoniRaphson iterai tion we illustrate this by considering lidimensional and didimensional maps 121 Where are the cycles Q What if you choose a really bad initial condition and it doesn t converge TD Lee Well then you only have yourself to blame TD Lee Ergodic exploration of recurrences that we turn to now sometimes per forms admirably well in getting us started In the Rossler ow example we sketched the attractors by running a long chaotic trajectory and noted that the attractors are very thin but otherwise the return maps that we plotted were disquieting 7 Fig 33 did not appear to be a litoil map In this section we show how to use such information to approximately locate cycles In the remainder of this chapter and in Chapter 7 we shall learn how to turn such guesses into highly accurate cycles Example 121 Rossler attractor G Simon and P Cvitanovic Run a long simulation of the Rossler ow ft plot a Poincare section as in Fig 31 and extract the corresponding Poincare return map P as in Fig 33 Luck is with us the Fig 121 a return map y gt P1yz looks much like a parabola so we take the unimodal map symbolic dynamics Section 1021 as our guess for the covering dynamics Strictly speaking the attractor is fractal but for all practical purposes the return map is lidimensional your printer will need a resolution better than 1014 dots per inch to start resolving its structure cycles 7 ZZapKZUU7 ChaasBook mgversionll a ZAug212007 121 WHERE ARE THE CYCLES 159 c xquot 7 d e Fig 121 a y 4 P1y 2 return map for m 0 y gt 0 Poincare section of the Rossler ow Fig 23 b The Trcycle found by taking the xed point ykn yk together with the xed point of the z a 2 return map not shown an initial guess 0 110 20 for the NewtonrRaphson search c yk3 P13yk 2k the third iterate ofPoincare return map 31 together with the corresponding plot for zk3 P2 yk 2k is used to pick starting guesses for the NewtonrRaphson searches for the two Srcycles d them cycle and e the m cycle C Simon Periodic points of a prime cycle p of cycle length mg for the x 0 y gt 0 Poincare section ofthe Rossler ow Fig 23 are xed points y z Pquot y z of the nth Poincare return map Using the xed point yk1 yk in Fig 121 a together with the simulta7 neous xed point of the z A P1y z return map not shown as a starting guess 0 110 zo for the NewtoniRaphson search for the cycle p with symi bolic dynamics label T we nd the cycle Fig 121 b with the Poincare sec tion point 0 ypzp period Tp expanding marginal contracting stability eigenvalues APVhApym A and Lyapunov exponents ARE Apym A 0 509176832 12997319 588108845586 7240395353110 14 129 X 10 014914155610447544 122 zyzgt T1 A1EA1mA1c A1571m71c Ticycle The NewtoniRaphson method that we used is described in Section 125 As an example ofa search for longer cycles we use yk3 P13yk zk the third iterate of Poincare return map 31 plotted in Fig 121 c together with a corresponding plot for n3 fagk zk to pick starting guesses for the NewtoniRaphson searches for the two Bicycles plotted in Fig 121 d e For a listing of the short cycles of the Rossler ow consult Table 7 The numerical evidence suggests but a proof is lacking that all cycles that comprise the strange attractor of the Rossler system are hyperbolic each with an expanding eigenvalue MA gt 1 a contracting eigenvalue lAcl lt l and a marginal eigenvalue lAml 1 corresponding to displacements along the direction of the ow For the Rossler system the contracting eigenvalues turn out to be insanely contracting a factor of 5 32 per one paricourse of the attractor so their nu ChaasBook mgvexsmnll a ZAug212007 cycles 7 ZZaptZOO M 127 page 189 0 0 02 04 06 08 1 Fig 122 The inverse time path to the y cycle of the logistic map fz 4z1 ix from an initial guess of m 02 At each inverse iteration we chose the 0 respecr tively 1 branch 1210 page 169 i M 75 wk p w w 20 25 30 735 0246 81012141618 20 Fig 123 Convergence of Newton s inverse iteration The error after n iterations searching for the Wicycle of the logistic map fz 4m1 7 z with an initial starting guess of 1 02z2 08 yraxis is log10 of the error The difference between the ex ponential convergence of the inverse iterr ation method and the superrexponential convergence of Newton s method is drar matic 160 CHAPTER 12 FIXED POINTS AND HOW TO GET THEM merical determination is quite dif cult Fortunately they are irrelevant for all practical purposes the strange attractor of the Rossler system is lidimensional a very good realization of a horseshoe template 122 One dimensional mappings 1221 Inverse iteration Let us rst consider a very simple method to nd unstable cycles of a lidimensional map such as the logistic map Unstable cycles of lid maps are attracting cycles of the inverse map The inverse map is not single valued so at each backward iteration we have a choice ofbranch to make By choosing branch according to the symbolic dynamics of the cycle we are trying to nd we will automatically converge to the desired cycle The rate of convergence is given by the stability of the cycle ie the convergence is exponentially fast Figure 122 shows such path to the Wicycle of the logistic map The method ofinverse iteration is ne for nding cycles for 1d maps and some 27d systems such as the repeller of Exercise 1210 It is not particularly fast especially if the inverse map is not known analyti ically However it completely fails for higher dimensional systems where we have both stable and unstable directions Inverse iteration will exchange these but we will still be left with both stable and uh stable directions The best strategy is to directly attack the problem of nding solutions of fT 1 1222 Newton s method Newton s method for determining a zero 1 of a function Fz of one variable is based on a linearization around a starting guess 10 FW FI0FIo110 123 An approximate solution 11 of Fz 0 is 11 10 FIoFIo 124 The approximate solution can then be used as a new starting guess in an iterative process A xed point of a map f is a solution to Fz z 7 fx 0 We determine 1 by iterating 9Im71 Emil Fzm71FIm71 Emil meil flt1m71A Em 125 Provided that the xed point is not marginally stable f z y l at the xed point z a xed point of f is a superistable xed point of the NewtoniRaphson map 9 g 0 and with a suf ciently good initial guess the NewtoniRaphson iteration will converge superiexponentially fast cycles 7 ZZaptZOO ChaasBook mgvexsmnll a ZAug212007 123 MULTIPOINT SHOOTING METHOD 161 To illustrate the ef ciency of the Newton s method we compare it to the inverse iteration method in Fig 123 Newton s method wins hands down the number of signi cant digits of the accuracy of I estimate doubles with each iteration In order to avoid jumping too far from the desired 1 see Fig 124 one often initiates the search by the damped Newton s method FIm FIm Armxm1izmi AT 0ltAT 17 takes small A7 steps at the beginning reinstating to the full A7 1 jumps only when suf ciently close to the desired 1 123 Multipoint shooting method Periodic orbits oflength n are xed points of f so in principle we could use the simple Newton s method described above to nd them How ever this is not an optimal strategy f will be a highly oscillating funce tion with perhaps as many as 2 or more closely spaced xed points and nding a speci c periodic point for example one with a given syme bolic sequence requires a very good starting guess For binary symbolic dynamics we must expect to improve the accuracy of our initial guesses by at least a factor of 2 to nd orbits of length n A better alternative is the multipoint shooting method While it might very hard to give a precise initial point guess for a long periodic orbit if our guesses are informed by a good state space partition a rough guess for each point along the desired trajectory might suf ce as for the individual short trajectory segments the errors have no time to explode exponentially A cycle of length n is a zero of the nedimensional vector function F 11 11 flt1n Fa F 12 12 f11 In In flt1n71 The relations between the temporal symbol sequences and the spatial layout of the topologically distinct regions of the state space discussed in Chapter 10 enable us to guess location of a series of periodic points along a cycle Armed with such informed initial guesses we can initiate a NewtoneRaphson iteration The iteration in the Newton s method now takes the form 0 126 cycles 7 ZZapKZUU7 d gone 7 z we ChaasBook mgversionll a zAug212007 Fx Fig 124 Newton method bad initial guess za leads to the Newton estimate zltb1 far away from the desired zero of Fwy Sequence Wm mm1 Y starting with a good guess converges superrexponentially to mquot The method diverges ifit iterates into the basin of at traction ofa local minimum we 162 CHAPTER 12 FIXED POINTS AND HOW TO GET THEM where FQ is an n X 71 matrix 1 f W f m 1 l 7 1 1771 127 This matrix can easily be inverted numerically by rst eliminating the elements below the diagonal This creates non7zero elements in the nth column We eliminate these and are done Let us take it step by step for a period 3 cycle Initially the setup for the Newton step looks like this 1 0 ifzg 61 7F1 7f zl 1 0 62 iFg 128 0 ifzg 1 63 7F3 where 61 zg71i is the correction of our guess for a solution and where Fi xi 7 zz1 First we eliminate the below diagonal elements by adding f zl times the rst row to the second row then adding f 12 times the second row to the third row We then have 1 0 7 f 13 61 0 1 ff gt a gt 0 0 17 f zgf zlf zg 63 129 7 F1 F2 11 F1 7F3 7 f z2F2 7 12f I1F1 The next step is to invert the last element in the diagonal ie divide the third row by l 7 f zgf zlf zg It is clear that if this element is zero at the periodic orbit this step might lead to problems In many cases this will just mean a slower convergence but it might throw the Newton iteration completely off We note that f zgf zlf xg is the stability of the cycle when the Newton iteration has converged and that this therefore is not a good method to nd marginally stable cycles We now have 1 0 fIs 51 0 1 f 11f 13 52 0 0 1 53 1 1210 F lt F27 f IiF1 FS f12F2 f12fTlFi lifTzVT fTs Finally we add 1 13 times the third row to the rst row and f 11f zg times the third row to the second row On the left hand side the matrix is now the unit matrix on the right hand side we have the corrections to our initial guess for the cycle ie we have gone through one step of the Newton iteration scheme cycles 7 ZZapKZUU7 ChaasBook mgversmnll a ZAug212007 124 DiDIMENSIONAL MAPPINGS 163 When one sets up the Newton iteration on the computer it is not necessary to write the left hand side as a matrix All one needs is a vector containing the f a vector containing the n th column that is the cumulative product of the f zi s and a vector containing the right hand side After the iteration the vector containing the right hand side should be the correction to the initial guess E 1 2 1 168 1 Page 124 d dimensional mappings F Christiansen Armed with symbolic dynamics informed initial guesses we can utilize the NewtoniRaphson iteration in didimensions as well 1241 Newton s method for didimensional mappings Newton s method for lidimensional mappings is easily extended to higher dimensions In this case f xi is a d X d matrix Fz is then an nd gtlt nd matrix In each ofthe steps that we went through above we are then manipulating d rows of the left hand side matrix Remember that matrices do not commute 7 always multiply from the left In the inversion of the n th element of the diagonal we are inverting a d X d matrix 1 7 which can be done if none of the eigenvalues of H f ri equals 1 ie the cycle must not have any marginally stable directions Some didimensional mappings such as the Henon map 315 can be written as lidimensional time delay mappings of the form an fzi1zi2 1 zzid 1211 In this case Fz is an nxn matrix as in the case ofusual lidimensional maps but with nonizero matrix elements on d offidiagonals In the elimination of these offidiagonal elements the last d columns of the ma trix will become nonezero and in the nal cleaning of the diagonal we will need to invert a d X d matrix In this respect nothing is gained nu merically by looking at such maps as lidimensional time delay maps 125 Flows F Christiansen Further complications arise for ows due to the fact that for a perii odic orbit the stability eigenvalue corresponding to the ow direction of necessity equals unity the separation of any two points along a cycle remains unchanged after a completion ofthe cycle More unit eigenvali ues can arise if the ow satis es conservation laws such as the energy invariance for Hamiltonian systems We now show how such problems are solved by increasing the number of xed point conditions Section 5 21 ChaasBook mgversionll a ZAug212007 cycles 7 ZZapKZUU7 164 CHAPTER 12 FIXED POINTS AND HOW TO GET THEM 1251 Newton s method for ows A ow is equivalent to a mapping in the sense that one can reduce the ow to a mapping on the Poincar surface of section An autonomous ow 26 is given as ivz7 1212 The corresponding fundamental matrix M 432 is obtained by inte7 grating the linearized equation 49 811i M AM7 14171 78 along the trajectory The ow and the corresponding fundamental ma7 trix are integrated simultaneously by the same numerical routine lnte7 grating an initial condition on the Poincar surface until a later crossing of the same and linearizing around the ow we can write fz mfzMz7z 1213 Notice here that even though all of z z and fx are on the Poincar surface fz is usually not The reason for this is that M corresponds to a speci c integration time and has no explicit relation to the arbitrary choice of Poincar section This will become important in the extended Newton s method described below To nd a xed point of the ow near a starting guess I we must solve the linearized equation lt17 MW 7 z 7 7ltz 7 fltzgtgt 7 7Fltzgt 1214 where fx corresponds to integrating from one intersection ofthe Poincar surface to another and M is integrated accordingly Here we run into problems with the direction along the ow since 7 as shown in Sec7 tion 521 7 this corresponds to a unit eigenvector of M The matrix 1 7 M does therefore not have full rank A related problem is that the solution 1 of 1214 is not guaranteed to be in the Poincar sur7 face of section The two problems are solved simultaneously by adding a small vector along the ow plus an extra equation demanding that I be in the Poincar surface Let us for the sake of simplicity assume that the Poincar surface is a hyper7plane ie it is given by the linear equation I710a07 1215 where a is a vector normal to the Poincar section and x0 is any point in the Poincar section 1214 then becomes lt2Mvlta gtgtltxgrgt7lt ltzgtgt The last row in this equation ensures that I will be in the surface of section and the addition of vr6T a small vector along the direction cycles7ZZap12007 ChaasBook mgversmnll a ZAug212007 125 FLOWS 165 of the ow ensures that such an I can be found at least ifz is suf ciently close to a solution ie to a xed point of f To illustrate this little trick let us take a particularly simple exam ple consider a 3d ow with the 17y707plane as Poincare section Let all trajectories cross the Poincare section perpendicularly ie with v 07 07 1 which means that the marginally stable direction is also perpendicular to the Poincare section Furthermore let the unstable die rection be parallel to the ziaxis and the stable direction be parallel to the yiaxis In this case the Newton setup looks as follows 17A 0 0 0 61 7E 0 17A 0 0 6y 7 iFy 0 0 0 v 6 7F 12m 0 0 1 0 6t 0 If you consider only the upperileft 3 X 3 matrix which is what we would have without the extra constraints that we have introduced then this matrix is clearly not invertible and the equation does not have a unique solution However the full 4 X 4 matrix is invertible as det vzdet 1 7 ML where Mi is the monodromy matrix for a surface of section transverse to the orbit see for ex 7 For periodic orbits 1216 generalizes in the same way as 127 but with 71 additional equations 7 one for each point on the Poincare sure face The Newton setup looks like this 1 4 711 1 v1 61 iFl 1 62 7F2 n71 1 Un 6n Fn a 5751 0 0 67in 0 a Solving this equation resembles the corresponding task for maps How ever in the process we will need to invert an d1n gtlt d1n matrix rather than a d X d matrix The task changes with the length of the cyi c e This method can be extended to take care of the same kind of prob lems if other eigenvalues of the fundamental matrix equal 1 This hapi pens if the ow has an invariant of motion the most obvious example being energy conservation in Hamiltonian systems In this case we add an extra equation for z to be on the energy shell plus and extra vari7 able corresponding to adding a small vector along the gradient of the Hamiltonian We then have to solve zuz 7z7fz 17M 711 WM 6t 0 1218 lt a 0 0 6E gt lt 0 ChaasBook mgversmnll a ZAug212007 cycles 7 ZZapKZUU7 166 CHAPTER 12 FIXED POINTS AND HOW TO GET THEM Fig 125 Illustration of the optimal Poincare surface The original surface y 0 yields a large distance x 7 fz for the Newton iteration A much better choice is y 07 simultaneously with Hz 7Hz0 1219 The last equation is nonlinear It is often best to treat this equation separately and solve it in each Newton step This might mean putting in an additional Newton routine to solve the single step of 1218 and 1219 together One might be tempted to linearize 1219 and put it into 1218 to do the two different Newton routines simultaneously but this will not guarantee a solution on the energy shell In fact it may not even be possible to nd any solution of the combined linearized equations if the initial guess is not very good 1252 Newton s method with optimal surface of section E Christiansen In some systems it might be hard to nd a good starting guess for a xed point something that could happen if the topology andor the symbolic dynamics of the ow is not well understood By changing the Poincare section one might get a better initial guess in the sense that z and fx are closer together In Fig 1252 there is an illustration of this The gure shows a Poincare section y 0 an initial guess I the corresponding fx and pieces of the trajectory near these two points If the Newton iteration does not converge for the initial guess I we might have to work very hard to nd a better guess particularly if this is in a high7dimensional system high7dimensional might in this context mean a Hamiltonian system with 3 degrees of freedom But clearly we could easily have a much better guess by simply shifting the Poincare section to y 07 where the distance I 7 fx would be much smaller Naturally one cannot see by eye the best surface in higher dimensional systems The way to proceed is as follows We want to have a minimal distance between our initial guess I and the image of this We therefore integrate the ow looking for a minimum in the distance dt If 7 dt is now a minimum with respect to variations in f x but not necessarily with respect to I We therefore integrate I either forward or backward in time Doing this we mini7 cycles 7 ZZapKZUU7 ChaasBook mgversionll a ZAug212007 125 FLOWS 167 mile d with respect to I but now it is no longer minimal with respect to f z We therefore repeat the steps alternating between correcting z and f z In most cases this process converges quite rapidly The result is a trajectory for which the vector 7 1 connecting the two end points is perpendicular to the ow at both points We can now choose to de ne a Poincare surface of section as the hyperplane that goes through I and is normal to the ow at I In other words the sure face of section is determined by 1 7 I vz 0 1220 Note that fx lies on this surface This surface of section is optimal in the sense that a close return on the surface is a local minimum of the distance between I and f z But more importantly the part of the stability matrix that describes linearization perpendicular to the ow is exactly the stability of the ow in the surface of section when fx is close to I In this method the Poincare surface changes with each iteration of the Newton scheme Should we later want to put the xed point on a speci c Poincare surface it will only be a matter of moving along the trajectory Summary There is no general computational algorithm that is guaranteed to nd all solutions up to a given period Tmax to the periodic orbit condition f TI f r7 T gt 0 for a general ow or mapping Due to the exponential divergence of nearby trajectories in chaotic dynamical systems direct solution of the periodic orbit condition can be numerically very unstab e A prerequisite for a systematic and complete cycle search is a good but hard to come by understanding of the topology of the ow Usui ally one starts by 7 possibly analytic 7 determination of the equilibria of the ow Their locations stabilities stability eigenvectors and in variant manifolds offer skeletal information about the topology of the ow Next step is numerical longitime evolution of typical trajeci tories of the dynamical system under investigation Such numerical experiments build up the natural measure and reveal regions most frequently visited The periodic orbit searches can then be initial ized by taking nearly recurring orbit segments and deforming them into a closed orbits With a suf ciently good initial guess the Newton Raphson formula 1216 17M vz 61 7 a 0 6T 7 0 yields improved estimate 1 1 61 T T6T lteration then yields the period T and the location of a periodic point I in the Poincare ChaasBook mgversmnll a ZAug212007 cycles 7 ZZapKZUU7 j Section 7 a Chapter 7 168 Exercises surface 117 7 10 a 07 where a is a vector normal to the Poincare section at 10 The problem one faces with highidimensional ows is that their topoli ogy is hard to visualize and that even with a decent starting guess for a point on a periodic orbit methods like the NewtoniRaphson method are likely to fail Methods that start with initial guesses for a number of points along the cycle such as the multipoint shooting method of Section 123 are more robust The relaxation or variational methods take this strategy to its logical extreme and start by a guess of not a few points along a periodic orbit but a guess of the entire orbit As these methods are intimately related to variational principles and path integrals we postpone their introduction to Chapter 7 Further reading Pieceiwise linear maps The Lozi map 317 is linear and multiplication and inversion 100000 5 of cycles can be easily computed by 2x2 matrix Exercises 121 122 123 Cycles of the Ulam map Test your cycle searching routines by computing a bunch of short cycles and their stabilities for the Ulam map 2 4961 790 1221 Cycles stabilities for the Ulam map exact In Exercise 121 you should have observed that the numerical results for the cycle stability eigenvalues 438 are exceptionally simple the stability eigen7 value of the 0 0 xed point is 4 while the eigen7 value of any other nicycle is i2quot Prove this Hint the Ulam map can be conjugated to the tent map 106 This problem is perhaps too hard but give it a try 7 the answer is in many introductory books on nolinear dynamics Stability of billiard cycles 0 few simple cycles Compute stabilities a A simple scattering billiard is the twoidisk bile liard It consists of a disk of radius one ceni tered at the origin and another disk of unit exaCycles r mbzoov radius located at L 2 Find all periodic or bits for this system and compute their stabilii ties You might have done this already in Ex ercise 12 at least now you will be able to see where you went wrong when you knew nothi ing about cycles and their extraction Find all periodic orbits and stabilities for a bile liard ball bouncing between the diagonal y x and one ofthe hyperbola branches y 12 124 Cycle stability Add to the pinball simulator of Exercise 81 a routine that evaluates the expanding eigenvalue for a given cycle 125 Pinball cycles Determine the stability and length ChaosBook mgversmnll a 2 Aug 212007 125 126 127 128 129 1210 REFERENCES of all fundamental domain prime cycles of the bi nary symbol string lengths up to 5 or longer for R a 6 37disk pinball NewtoniRaphson method Implement the NewtoniRaphson method in 2d and apply it to de termination of pinball cycles Rossler system cycles continuation of Exeri c1se Determine all cycles up to 5 Poincare sections ref turns for the Rossler system 215 as well as their stabilities Hint implement 1216 the multipoint shooting methods for ows you can crossicheck your shortest cycles against the ones listed in Ta ble 7 Cycle stability helium Add to the helium intei grator of Exercise 210 a routine that evaluates the expanding eigenvalue for a given cyc e Colinear helium cycles Determine the stability and length of all fundamental domain prime cyi cles up to symbol sequence length 5 or longer for collinear helium of Fig 77 Uniqueness of unstable cycles Prove at there exists only one 37disk prime cycle for a given nite admissible prime cycle symbol string 1212 Hints look at the Poincare section mappings can you show that there is exponential contraction to a References 169 unique periodic point With a given itinerary Exeri cise 7 might be helpful in this effort Inverse iteration method for a Hamiltonian ref peller Consider the Henon map 315 for areaipreserving Hamiltonian parameter value b 71 The coordinates of a periodic orbit oflength 71p satisfy the equation 1pwi1 xpi71liaxyi i 1 W 71p 1222 With the periodic boundary condition Ipp p n Verify that the itineraries and the stabilities of the short periodic orbits for the Henon repeller 1222 at a 6 are as listed in Table 7 Hint you can use any cycleisearching routine you Wish but for the complete repeller case all binary sequences are realized the cycles can be evalu7 ated simply by inverse iteration using the inverse of 1222 N iw39 Spi Here SN are the signs of the corresponding cycle point coordinates SN xwxw G Vattay Center ofmass puzzle Wh is the center of mass listed in Table 7 a simple rational every so ten 1 M Baranger and KTR Davies Ann Physics 177 330 1987 2 ED Mestel and l Percival Physica D 24 172 1987 Q Chen D 3 4 Meiss and l Percival Physica D 29 143 1987 nd Helleman et all Fourier series methods M Greene 1 Math Phys 20 1183 1979 5 HE Nusse and Yorke quotA procedure for nding numerical trai jectories on chaotic saddles Physica D 36 137 1989 6 DP Lathrop and E Kostelich quotCharacterization of an experi mental strange attractor by periodic orbits 7 T E Huston KTR Davies and M Baranger Chaos 2 215 1991 8 M Brack R K Bhaduri J Law and M V N Murthy Phys Rev Lett 70568 1993 9 Z Gills C lwata R Roy LB Scwartz and l Triandaf Tracking Unstable Steady States Extending the Stability Regime of a Multii mode Laser System Phys Rev Lett 69 3169 1992 10 NJ Balmforth P Cvitanovi GR lerley EA Spiegel and G Vate tay Advection of vector elds by chaotic ows Stochastic Prof ChaasBook mgversmnll 9 ZAug212007 refsCycles 7 2761222004 Spectral determinants It seems very pretty 1 she said when she had nished it but it s rather hard to understand You see she didn t like to con7 fess even to herself that she couldn t make it out at all Some7 how it seems to ll my head with ideas 7 only I don t exactly know what they are 1 Lewis Carroll Through the Looking Glass The problem with the trace formulas 1610 1623 and 1628 is that they diverge at z e40 respectively 3 so ie precisely where one would like to use them While this does not prevent numerical estima7 tion of some thermodynamic averages for iterated mappings in the case of the Gutzwiller trace formula of Chapter 7 this leads to a per7 plexing observation that crude estimates of the radius of convergence seem to put the entire physical spectrum out of reach We shall now cure this problem by thinking at no extra computational cost while traces and determinats are formally equivalent determinants are the tool of choice when it comes to computing spectra The idea is illus7 trated by Fig 113 Determinants tend to have larger analyticity do7 mains because if tr 1 7 zL 7 i In det 17 zL diverges at a partic7 ular value of 2 then det 1 7 zL might have an isolated zero there and a zero ofa function is easier to determine numerically than its poles 171 Spectral determinants for maps The eigenvalues 2k of a linear operator are given by the zeros of the determinant det17 2r Ha 7 zzk 171 k For nite matrices this is the characteristic determinant for operators this is the Hadamard representation of the spectral determinant sparing the reader from pondering possible regularization factors Consider rst the case of maps for which the evolution operator advances the densities by integer steps in time In this case we can use the formal matrix identity lndet1 7M tr ln17M 721nm 172 TL 171 Spectral determinants for maps 245 172 Spectral determinant for ows 246 173 Dynamical zeta functions 249 174 False zeros 252 175 Spectral determinants Vs dynamr ical zeta functions 252 176 All too many eigenvalues 254 Summa 255 Further reading 255 Exercises 256 References 258 E Chapter 77 W Chapter 739 Appendix 7 Z 4 1 page 67 Z46 CHAPTER 17 SPECTRAL DETERMINANTS to relate the spectral determinant of an evolution operator for a map to its traces 168 and hence to periodic orbits39 co Zn detlizL exp izjitr L n a 0 1 27217578 Ap exp 72W 173 p 771 p Going the other Way the trace formula 1610 can be recovered from the spectral determinant by taking a derivative L o Ulii izflndctuizc 174 2 1 7 2L 7 W fast track Section 17 2 p 246 Example 171 Spectral determinants of transfer operators 9 For a piecewiserlinear map 1517 with a nite Markov partition an explicit formula for the spectral determinant follows by substituting the trace formula 1611 into 173 to ti detlizll 17777 175 1 gt A3 M where ts zlASl The eigenvalues are necessarily the same as in 1612 which We already determined from the trace formula 1610 e exponential spacing of eigenvalues guarantees that the spectral deterr minant 175 is an entire function It is this property that generalizes to i 39 39 39 39 Markov pariminn 39 39 I i c as the tool of choice for evaluation of spectra 172 Spectral determinant for ows an analogue of the ArtinrMazur zeta function for diffeor morphisms seems quite remote for ows However we will mention a wild idea in this direction 1 de ne 1v to be u r m iuuiei and r function Zs to be the in nite product 2a H f1 1 7 expl7395quot Stephen Smale Differentiable Dynamical Systems eel lBaprI S CbosBaukorgvuswnnBZAug 2i m 172 SPECTRAL DETERMINANT FOR FLOWS 247 We write the formula for the spectral determinant for ows by anal7 ogy to 173 dets7A exp det17M erasArm 176 and then check that the trace formula 1623 is the logarithmic deriva7 tive of the spectral determinant tr A 1 5 d3 d lndets7A 177 With 2 set to z e4 as in 1624 the spectral determinant 176 has the same form for both maps and ows We refer to 176 as spectral determinant as the spectrum of the operator A is given by the zeros of dets7A0 178 We now note that the 7 sum in 176 is close in form to the expansion of a logarithm This observation enables us to recast the spectral deter7 minant into an in nite product over periodic orbits as follows Let Mp be the p7cycle d X d transverse fundamental matrix with eigenvalues A11 A1 2 AW Expanding the expanding eigenvalue factors 11 7 1Ape and the contracting eigenvalue factors 11 7 Aim in 164 as geometric series substituting back into 176 and resum7 ming the logarithms we nd that the spectral determinant is formally given by the in nite product det s 7 A 117 18 10 H o 1 l l l H 1 it AZ7Igt 1AZZ 2 39 quotA1771 77 p tp273v 1 H 16145 k1 kg 1 AzulAz 39 39 39APgt 3 1 e AP75TPZ7LP 1M gt 179 1710 In such formulas tp is a weight associated with the p cycle letter t refers to the local trace evaluated along the p cycle trajectory and the index p runs through all distinct prime cycles When convenient we inserts the 2 factor into cycle weights as a formal parameter which keeps track ofthe topological cycle lengths These factors will assists us in expanding zeta functions and determinants eventually we shall set 2 1 The subscripts e7 5 indicate that there are e expanding eigenval7 ues and c contracting eigenvalues The observable whose average we wish to compute contributes through the A term in the p cycle mul7 tiplicative weight e 39AP By its de nition 151 the weight for maps is a product along the cycle points ChaasBook mgversmnll a ZAug212007 77171 EA H eawm j0 c1217 lBaprZOOE g Chapter 18 2 1 7 page 7 248 CHAPTER 17 SPECTRAL DETERMINANTS and the weight for ows is an exponential of the integral 155 along the cycle TP GAP exp az7 d7 gt 0 This formula is correct for scalar weighting functions more general ma trix valued weights require a timeiordering prescription as in the fun damental matrix of Section 41 Example 172 Expanding 17d map a For expanding 17d mappings the spectral determinant 179 takes the form det17z w17tAk 17LP 1711 Elly P P p mp1 0 Example 173 Twoidegree of freedom Hamiltonian ows For a Zidegree of freedom Hamiltonian ows the energy conservation elimii nates on phase space variable and restriction to a Poincare section eliminates the marginal longitudinal eigenvalue A 1 so a periodic orbit of Zidegree of freedom hyperbolic Hamiltonian ow has one expanding transverse eigen7 value A 1A1 gt 1 and one contracting transverse eigenvalue 1 A The weight in 164 is expanded as follows k1 A5 1712 1 1 1 ldet1 711ml 1111107 1A2 W The spectral determinant exponent can be resummed co 1 C ApisTPM ltgtltgt e AfsTP 7 ldetui Mm g 1gt1 glt1 1AM gt and the spectral determinant for a Zidimensional hyperbolic Hamiltonian ow rewritten as an in nite product over prime cycles dets 7A H 10 01171pA Qk1 p k0 1713 Now we are nally poised to deal with the problem posed at the beginning of Chapter 16 how do we actually evaluate the averages in troduced in Section 151 The eigenvalues of the dynamical averaging evolution operator are given by the values of s for which the spectral determinant 176 ofthe evolution operator 1523 vanishes If we can compute the leading eigenvalue so 5 and its derivatives we are done Unfortunately the in nite product formula 179 is no more than a shorthand notation for the periodic orbit weights contributing to the spectral determinant more work will be needed to bring such formui las into a tractable form This shall be accomplished in Chapter 18 but here it is natural to introduce still another variant of a determinant the dynamical zeta function c1217 lBaprZOOE ChaosBook mgvasmu a 2 Aug 212007 173 DYNAMICAL ZETA FUNCTIONS 249 173 Dynamical zeta functions It follows from Section 1611 that if one is interested only in the leading eigenvalue of U the size of the p cycle neighborhood can be approxi7 mated by IIAp T the dominant term in the 7T1 t A 00 limit where Ap He Ana is the product of the expanding eigenvalues of the fun damental matrix Mp With this replacement the spectral determinant 176 is replaced by the dynamical zeta filnction 14 exp 722 1714 17 71 that we have already derived heuristically in Section 152 Resumming the logarithms using ET 727 71n1 7 11 we obtain the Eulerproduct representation of the dynamical zeta function 14 Haiti 1715 17 In order to simplify the notation we usually omit the explicit depen7 dence of 14 tp on 2 s 5 whenever the dependence is clear from the context The approximate trace formula 1628 plays the same role Visiaivis the dynamical zeta function 177 d Tt Fsglnlt 1zlf 1716 p 17 as the exact trace formula 1623 plays Visiaivis the spectral determine ant 176 The heuristically derived dynamical zeta function of Sec tion 152 now reiemerges as the 14 00z part of the exact spectral determinant other factors in the in nite product 179 affect the non leading eigenvalues of L In summary the dynamical zeta function 1715 associated with the ow 1 is de ned as the product over all prime cycles p The quan7 tities T1 711 and A1 denote the period topological length and product of the expanding stability eigenvalues of prime cycle p A17 is the inte7 grated observable az evaluated on a single traversal of cycle p see 155 s is a variable dual to the time t z is dual to the discrete topoi logical time n and tpzs denotes the local trace over the cycle p We have included the factor 2 in the de nition of the cycle weight in order to keep track of the number of times a cycle traverses the surface of section The dynamical zeta function is useful because the term 1lts0 1717 when 3 50 Here so is the leading eigenvalue of U e which is often all that is necessary for application of this equation The above argument completes our derivation of the trace and determinant for mulas for classical chaotic ows In chapters that follow we shall make these formulas tangible by working out a series of simple examples ChaasBook mgversmnll a ZAug212007 c1217 lBaprZOOE 2 17 7 page 25 W Chapter 18 250 CHAPTER 17 SPECTRAL DETERMINANTS The remainder of this chapter offers examples of zeta functions W fast track Chapter 18 p 261 1731 A contour integral formulation The following observation is sometimes useful in particular for zeta functions with richer analytic structure than just zeros and poles as in the case of intermittency Chapter 7 I m the trace sum 1626 can be expressed in terms of the dynamical zeta function 1715 11 13lt1 7 1718 as a contour integral 1 d r i in 71 1 d 2mg dim w z where a small contour 7 encircles the origin in negative clockwise direction If the contour is small enough ie it lies inside the unit circle 1 we may write the logarithmic derivative of 42 as convergent sum over all periodic orbits Integrals and sums can be interchanged the integrals can be solved term by term and the trace formula 1626 is recovered For hyperbolic maps cycle expansions or ther techniques provide an analytical continuation of the dynamical zeta function beyond the leading zero we may therefore deform the original contour into a larger circle with radius R which encircles both poles and zeros of C as depicted in Fig 171 Residue calculus turns this into a sum over the zeros 2a and poles 23 of the dynamical zeta function that is 1719 poles 1 1 1 d if if in nilogg l zeros I 1720 leKR where the last term gives a contribution from a large circle 7 It would be a miracle if you still remebered this but in Section 143 we inter preted IL as fraction of survivors after 71 bounces and de ned the es cape rate 7 as the rate of the nd exponential decay of Pn We now see that this exponential decay is dominated by the leading zero or pole of 12 1732 Dynamical zeta functions for transfer operators 1 J Ruelle39 39 39 m H the topological zeta function 1321 to a function that assigns different am lBaprIElS bosBaukorgvaswnllBZAug 21 m 173 DYNAMICAL ZETA FUNCTIONS 251 weights to different cycles 42 exp 2 711 Here we sum over all periodic points I of period n and 91 is any matrix valued weighting function where the weight evaluated multii plicatively along the trajectory of 11 By the chain rule 438 the stability of any nicycle of a lid map is given by Ap 111 f zi so the 17d map cycle stability is the sime plest example of a multiplicative cycle weight llf zi and indeed 7 via the PerroniFrobenius evolution operator 149 7 the histor7 ical motivation for Ruelle s more abstract construction In particular for a piecewiseilinear map with a nite Markov partii tion such as the map of Example 141 the dynamical zeta function is given by a nite polynomial a straightforward generalization of the topological transition matrix determinant 102 As explained in Sec tion 133 for a nite N X N dimensional matrix the determinant is given by 2 tr 9fjzz xeFixfquot J N Hl 7 tp Zzncn p 711 where on is given by the sum over all noniselfiintersecting closed paths of length n together with products of all noniintersecting closed paths of total length n Example 174 A piecewise linear repeller Due to piecewise linearity the stability of any nicycle of the piecewise lin ear repeller 1517 factorizes as A5152 5 A6quotA m where m is the total number of times the letter 5739 0 appears in the p symbol sequence so the traces in the sum 1628 take the particularly simple form tr Tquot FT L le1 1M The dynamical zeta function 1714 evaluated by resumming the traces 1 z1z1Aolz1Ally 1721 is indeed the determinant det17 zT of the transfer operator 1519 which is almost as simple as the topological zeta function 1325 More generally piecewiseilinear approximations to dynamical sys7 tems yield polynomial or rational polynomial cycle expansions pro vided that the symbolic dynamics is a subshift of nite type We see that the exponential proliferation of cycles so dreaded by quantum chaologians is a bogus anxiety we are dealing with expo nentially many cycles of increasing length and instability but all that really matters in this example are the stabilities of the two xed points Clearly the information carried by the in nity of longer cycles is highly redundant we shall learn in Chapter 18 how to exploit this redundancy systematically ChaasBook mgversmnll a ZAug212007 c1217 lBaprZOOE 16 2 page 243 17 3 page 257 Section10 5 252 CHAPTER 17 SPECTRAL DETERMINANTS 174 False zeros Compare 1721 with the Euler product 1715 For simplicity consider two equal scales A01 A11 e Our task is to determine the leading zero 2 e7 of the Euler product It is a novice error to assume that the in nite Euler product 1715 vanishes whenever one of its factors vanishes If that were true each factor 1 7 znPAp would yield 0 17 whim 1722 so the escape rate 7 would equal the stability exponent of a repulsive cycle one eigenvalue 7 71 for each prime cycle p This is false The exponentially growing number of cycles with growing period conspires to shift the zeros of the in nite product The correct formula follows from 1721 0 17 WW h 1112 1723 This particular formula for the escape rate is a special case of a general relation between escape rates Lyapunov exponents and entropies that is not yet included into this book Physically this means that the escape induced by the repulsion by each unstable xed point is diminished by the rate of backscatter from other repelling regions ie the entropy h the positive entropy of orbits shifts the false zeros z eAP of the Euler product 1715 to the true zero 2 e h 175 Spectral determinants vs dynamical zeta functions In Section 173 we derived the dynamical zeta function as an approx imation to the spectral determinant Here we relate dynamical zeta functions to spectral determinants exactly by showing that a dynami ical zeta function can be expressed as a ratio of products of spectral determinants The elementary identity for didimensional matrices 1 d 1 Wg hk WM 7 1724 inserted into the exponential representation 1714 of the dynamical zeta function relates the dynamical zeta function to weighted spectral determinants Example 175 Dynamical zeta function in terms of determinants led ma 5 For 17d maps the identity 17 1 7171ATX171A c1217 lBaprZOOE ChaosBook orgvzxsmnll a 2 Aug 212007 176 ALL TOO MANY EIGENVALUES 253 substituted into 1714 yields an expression for the dynamical zeta function for 1d maps as a ratio of two spectral determinants det1 7 z VC ma a 1725 where the cycle weight in Q1 is given by replacement tp 7gt tpAp As we shall see in Chapter 7 this establishes that for nice hyperbolic ows 1g is meromorphic with poles given by the zeros of det 17z 1 The dynamical zeta function and the spectral determinant have the same zeros although in exceptional circumstances some zeros of det 17z 1 might be cancelled by coincident zeros of det1 7 z m Hence even though we have derived the dynamical zeta function in Section 173 as an approximation to the spectral determinant the two contain the same spectral information Example 176 Dynamical zeta function in terms of determinants 27d Hamiltonian ma 5 For Z7dimensional Hamiltonian ows the above identity yields 1 l 2 W1A1lt171Agt21 2A1A det17z det17 z m det17z 1 This establishes that for nice 27d hyperbolic ows the dynamical zeta func7 tion is meromorphic 1g 1726 Example 177 Dynamical zeta functions for 2d Hamiltonian ows The relation 1726 is not particularly useful for our purposes Instead we insert the identity 1 1 if 1 1 1 171192 A171A2 A2171A2 into the exponential representation 1714 of 1Ck and obtain det1 z m m 1 Mm W det17 z k12 1727 Even though we have no guarantee that det 1 7 z w are entire we do know that the upper bound on the leading zeros of det 1 7 z ltk1 lies strictly below the leading zeros of det 1 7 z w and therefore we expect that for Z7dimensional Hamiltonian ows the dynamical zeta function 1gk generically has a double leading pole coinciding with the leading zero of the det1 7 z k1 spectral determinant This might fail if the poles and lead7 ing eigenvalues come in wrong order but we have not encountered such situations in our numerical investigations This result can also be stated as follows the theorem establishes that the spectral determinant 1713 is en7 tire and also implies that the poles in 1Ck must have the right multiplicities to cancel in the det17 z H 1C61 product ChaasBook mgvexsmnll a ZAug212007 c1217 lBaprZOOE 4W1 ASAT 72m 7m n a a a o 0 073 0 c Fig 173 The classical resonances at 1mm 1728 for a Zrdisk game of pin ball 254 CHAPTER 17 SPECTRAL DETERMINANTS 176 All too many eigenvalues What does the 27dimensional hyperbolic Hamiltonian ow spectral deti erminant 1713 tell us Consider one of the simplest conceive able hyperbolic ows the game ofpinball of Fig 172 consisting of two disks of equal size in a plane There is only one periodic orbit with the period T and expanding eigenvalue A given by elementary considera7 tions see Exercise 93 and the resonances det 3a 7 A 0 a 16 n plotted in Fig 173 30 7k1n2727 n e Z k e Zl multiplicity 1H1 1728 can be read off the spectral determinant 1713 for a single unstable cycle dets 7 A H 17 e STAlAkk1 1729 In the above A ln AlT is the cycle Lyapunov exponent For an open system the real part of the eigenvalue 50 gives the decay rate of ath eigenstate and the imaginary part gives the node number of the eigenstate The negative real part of 50 indicates that the resonance is unstable and the decay rate in this simple case zero entropy equals the cycle Lyapunov exponent Rapidly decaying eigenstates with large negative 9950 are not a prob lem but as there are eigenvalues arbitrarily far in the imaginary direc7 tion this might seem like all too many eigenvalues However they are necessary 7 we can check this by explicit computation of the right hand side of 1623 the trace formula for ows M8 2 k 1ek1ti27rntT 0 7 1 tT ix i27rn T k1ltWAkgt Z 6 k 1 AV T2060 41 0 6177 139 TgmiAHlelAT Q H o a H H M8 k 0 H M8 H o 1730 Hence the two sides of the trace formula 1623 are veri ed The for mula is ne for t gt 0 for t gt 0 however sides are divergent and need regularization The reason why such sums do not occur for maps is that for discrete time we work with the variable 2 as so an in nite strip along 35 maps into an annulus in the complex 2 plane and the Dirac delta sum in the above is replaced by the Kronecker delta sum in 168 In the case at hand there is only one time scale T and we couldjust as well det r leapizooa ChaosBook oxgvasimll a 2 Aug 212007 Q Fig 172 A game of two disks of equal si only periodic orbit Further reading 255 replace 3 by the variable 2 e ST In general a continuous time ow has an in nity of irrationally related cycle periods and the resonance arrays are more irregular cf Fig 181 Summary The eigenvalues of evolution operators are given by the zeros of corre sponding determinants and one way to evaluate determinants is to ex pand them in terms of traces using the matrix identity log det tr log Traces of evolution operators can be evaluated as integrals over Dirac delta functions and in this way the spectra of evolution operators are related to periodic orbits The spectral problem is now recast into a problem of determining zeros of either the spectral determinant 1 5 3Ar5Tp dets A explt ggwdetuiMzw or the leading zeros of the dynamical zeta function 1 Ha ftp tp ie 39A Tw p w The spectral determinant is the tool of choice in actual calculations as it has superior convergence properties this will be discussed in Chap ter 7 and is illustrated for example by Table 182 In practice both spectral determinants and dynamical zeta functions are preferable to trace formulas because they yield the eigenvalues more readily the main difference is that while a trace diverges at an eigenvalue and re quires extrapolation methods determinants vanish at 3 corresponding to an eigenvalue 5a and are analytic in s in an open neighborhood of 3a The critical step in the derivation of the periodic orbit formulas for spectral determinants and dynamical zeta functions is the hyperbole icity assumption 165 that no cycle stability eigenvalue is marginal AW 7 1 By dropping the prefactors in 14 we have given up on any possibility of recovering the precise distribution of the initial 1 return to the past is rendered moot by the chaotic mixing and the exponene tial growth of errors but in exchange we gain an effective description of the asymptotic behavior of the system The pleasant surprise to be demonstrated in Chapter 18 is that the in nite time behavior of an unstable system turns out to be as easy to determine as its short time behavior Further reading ChaasBook mgversmnll a ZAug212007 c1217 lBaprZOOE 256 Piecewise monotone maps A partial list of cases for which the transfer operator is well de ned the expand7 ing Holder case weighted subshifts of nite type expand7 ing differentiable case see Bowen 23 expanding holo7 morphic case see Ruelle 7 piecewise monotone maps of the interval see Hofbauer and Keller 15 and Baladi and Keller 18 Smale s wild idea Smale s wild idea quoted on page 246 was technically wrong because 1 the Sel7 berg zeta function yields the spectrum of a quantum me7 chanical Laplacian rather than the classical resonances 2 the spectral determinant weights are different from what Smale conjectured as the individual cycle weights also de7 pend on the stability of the cycle 3 the formula is not dimensionally correct as k is an integer and 5 represents inverse time Only for spaces of constant negative cur7 vature do all cycles have the same Lyapunov exponent A lnApTp In this case one can normalize time so that A l and the factors e STPA in 179 simplify to s skTP as intuited in Smale s quote on page 246 where l39y is the cycle period denoted here by Ty Nevertheless Smale s intuition was remarkably on the target Is this a generalization of the Fourier analysis Fourier analysis is a theory of the space lt gt eigenfunction duality for dynamics on a circle The way in which peri7 odic orbit theory generalizes Fourier analysis to nonlinear ows is discussed in Ref 4 a very readable introduction to the Selberg Zeta function Zeta functions antecedents For a function to be deserving of the appellation zeta function one ex7 pects it to have an Euler product representation 1715 and perhaps also satisfy a functional equation Various kinds of zeta functions are reviewed in Refs 8710 His7 torical antecedents of the dynamical zeta function are the xed7point counting functions introduced by Weil 11 Lefschetz 12 and Artin and Mazur 13 and the deter7 Exercises minants of transfer operators of statistical mechanics 25 In his review article Smale 22 already intuited by analogy to the Selberg Zeta function that the spectral det7 erminant is the right generalization for continuous time ows In dynamical systems theory dynamical zeta func7 tions arise naturally only for piecewise linear mappings for smooth ows the natural object for the study of clas7 sical and quantal spectra are the spectral determinants Ruelle derived the relation 173 between spectral deter7 minants and dynamical zeta functions but since he was motivated by the Artin7Mazur zeta function 1321 and the statistical mechanics analogy he did not consider the spectral determinant to be a more natural object than the dynamical zeta function This has been put right in papers on at traces 217 The nomenclature has not settled down yet what we call evolution operators here is elsewhere called trans7 fer operators 27 Perron7Frobenius operators 6 andor Ruelle7Araki operators Here we refer to kernels such as 1523 as evolution operators We follow Ruelle in usage of the term d n7 amical zeta function but elsewhere in the literature the function 1715 is often called the Ruelle zeta function Ruelle 28 points out that the corresponding transfer op7 erator T was never considered by either Perron or Frobe7 nius a more appropriate designation would be the Ruelle7 Araki operator Determinants similar to or identical with our spectral determinants are sometimes called Selberg Zetas Selberg7Smale zetas 8 functional determinants Fredholm determinants or even 7 to maximize confusion 7 dynamical zeta functions 14 A Fredholm determinant is a notion that applies only to trace class operators 7 as we consider here a somewhat wider class of operators we prefer to refer to their determinants loosely as spectral determinants Exercises 171 Escape rate for a 17d repeller numerically Con7 sider the quadratic map fzAzliz 1731 on the unit interval The trajectory of a point start7 ing in the unit interval either stays in the interval forever or after some iterate leaves the interval and eaneb Ammo diverges to minus in nity Estimate numerically the escape rate 7 the rate of exponential decay of the measure of points remaining in the unit interval for either A 92 or A 6 Remember to com7 pare your numerical estimate with the solution of the continuation of this exercise Exercise 182 172 Spectrum ofthe golden mean pruned map ChaosBook mgvexsmnll a 2 Aug 212007 Exercises medium 7 Exercise 136 continued a Determine an expression for tr Lquot the trace of powers of the Perron7Frobenius operator 1410 for the tent map of Exercise 136 b Show that the spectral determinant for the Perron7Frobenius operator is Z 22 det17z lt17W7Wgt k even Z 22 H 1 Ak1 Az z k odd 173 Dynamicalzeta functions easy a Evaluate in closed form the dynamical zeta function zquot 1 C z lt17 7 lt gt W for the piecewise7linear map 1517 with the left branch slope AD the right branch slope A1 257 in nite sum by the use of a logarithm Use the properties of in nite sums to develop a sensi7 ble de nition of in nite products If zquot is a root of the function F show that the in nite product diverges when evaluated at z How does one compute a root of a function represented as an in nite product 8 Let p be all prime cycles ofthe binary alphabet 01 Apply your de nition ofFz to the in7 nite product 1107173 F Fz Are the roots of the factors in the above prod7 uct the zeros of Fz7 Per Rosenqvist 176 Dynamical zeta functions as ratios of spectral medium Show that the zeta function 1 z 1 z ex 7 7 7 p p a Plt ggrm Sm 5 can be written as the ratio 1 z det17 z odet17 z m where det1 7 z s UPI 200 A0 A1 500 s10 zquotP1AplAZS 177 Contour integral for survival probability Per7 x x form explicitly the contour integral appearing in b What if there are four different slopes 1719 50075013510v and 511 mSteaq OfJUSt WOV 39WiTh 178 Dynamical zeta function for maps In this the prmmages 3me gap adJUSted SO matJUnF39 problem we will compare the dynamical zeta func7 Hons Of39branches 5007 501 and 5117510 map m tion and the spectral determinant Compute the ex7 The gap m one Brawn What Wm d The dyn39 act dynamical zeta function for the skew Ulam tent amical zeta function be map 1445 174 Dynamical zeta functions from Markov graphs Zn Extend Section 133 to evaluation of dynamical 1CZ H lt1 zeta functions for piecewise linear maps with nite P6P p MaTkOV graphs This generalizes The suns Of EX What are its roots Do they agree with those com7 ercise 173 puted in Exercise 1477 175 zeros 0f in nite Pmduas Demrmma m 0f 179 Dynamical zeta functions for Hamiltonian maps the quantities of interest by periodic orbits involves working with in nite product formulas a Consider the in nite product 1 Ha m k0 where the functions fk are suf ciently nice This in nite product can be converted into an ChaasBook mgversmnll a zAug212007 Starting from 1ltltsgt exp 722 for a Z7dimensional Hamiltonian map Using the equality 2 7 1 W20 2A1A gt ExExDetr AodZOOS 1710 258 show that 1g det1 7 det17 2det17 cm Exercises c Are the zeros of the terms in the product 5 7 ln p also the zeros of the Riemann C func7 tion If not why not In this expression det1 7 z k is the expansion 1711 Finite truncations easy Suppose we have a one7 d mens39 one gets by replacing tp 7gt tpA in the spectral determinant Riemann C function de ned as the sum 2 a Use factorization into primes to derive the Eu7 ler product representation 5H1771p5 The Riemann C function is SEC The dynamical zeta function Exercise 1715 is called a zeta function because it shares the form of the Euler product representation with the Riemann zeta function b Not trivialz For which complex values of s is e Riemann zeta sum convergent References 1onal system with complete binary dynam7 ics where the stability of each orbit is given by a simple multiplicative rule A AgP OAILP la p0 0111177 El 1111 so that for example A00101 AEAf a Compute the dynamical zeta function for this system perhaps by creating a transfer matrix analogous to 1519 with the right weights Compute the nite p truncations of the cycle expansion ie take the product only over the p up to given length with my g N and expand as a series in z 510 Do they agree If not how does the disagree7 ment depend on the truncation length N 7 1 D Ruelle Statistical Mechanics ThermodynamicFormalism Addison Wesley Reading MA 1978 2 D Ruelle Bull Amer Math Soc 78 988 1972 3 M Pollicott Invent Math 85 147 1986 4 HP McKean Comm Pure and Appl Math 25 225 1972 27 134 1974 5 W Parry and M Pollicott Ann Math 118 573 1983 6 Y Oono and Y Takahashi Progr Theor Phys 63 1804 1980 87 Chang and Wright Phys REV A 23 1419 1981 Y Takahashi and Y Oono Progr Theor Phys 71 851 1984 7 P Cvitanovic RE Rosenqvist HH Rugh and G Vattay CHAOS 3 619 1993 8 A Voros in Zeta Functions in Geometry Proceedings Tokyo 1990 eds N Kurokawa and T Sunada Advanced Studies in Pure Math ematics 21 Math Soc Japan Kinokuniya Tokyo 1992 p327 358 9 Kiyosi Ito ed Encyclopedic Dictionary of Mathematics MIT Press Cambridge 1987 10 NE Hurt Zeta functions and periodic orbit theory A review Results in Mathematics 23 55 Birkhauser Basel 1993 11 A Weil Numbers of solutions of equations in nite elds Bull Am 12313217 zasepzom ChaosBook mgversmnll 9 2 Aug 212007


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