Lectures on Chaotic s This page intentionally left blank https://doi.org/10.1090/amsip/028 AMS/IP Studies in Advanced Mathematics

Volume 28

Lectures on Chaotic Dynamical System s

Valentin Afraimovich and Sze-Bi Hsu

American Mathematical Society • International Press Shing-Tung Yau , Genera l Edito r

2000 Mathematics Subject Classification. Primar y 37-XX .

Library o f Congres s Cataloging-in-Publicatio n Dat a Afraimovich, V . S . (Valenti n Senderovich ) Lectures o n chaoti c dynamica l system s / Valenti n Afraimovic h an d Sze-B i Hsu . p. cm . — (AMS/I P studie s i n advance d mathematics , ISS N 1089-328 8 ; v. 28 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3168- 2 (alk . paper ) 1. Differentiabl e dynamica l systems . 2 . Chaoti c behavio r i n systems . I . Hsu , Sze-Bi , 1948 - II. Title . III . Series .

QA614.8.A385 200 2 514/.74—dc21 2002074423

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© 200 3 by the America n Mathematica l Societ y an d Internationa l Press . Al l right s reserved . The America n Mathematica l Societ y an d Internationa l Pres s retai n al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n this boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AM S hom e pag e a t http://www.ams.org / Visit th e Internationa l Pres s hom e pag e a t URL : http://www.intlpress.com / 10 9 8 7 6 5 4 3 2 1 0 8 0 7 0 6 0 5 0 4 0 3 Contents

1 Basi c Concept s 1 1.1 Th e Worl d o f the Observable s 1 1.2 Dynamica l System s 3 1.3 Dynamica l Chaos . Som e Definitions 6 1.4 System s with Dissipatio n 9 1.5 Strang e : Firs t Encounte r 2 6 1.6 Characteristic s o f Complexity o f Attractors 3 7 1.7 Mari e Theorem an d Taken s Definitio n 4 1

2 Zero-Dimensiona l Dynamic s 5 1 2.1 Symboli c Dynamic s 5 1 2.2 Application s o f the Bernoull i Schem e 5 5 2.3 "Two-sided " Bernoull i Shif t 6 2 2.4 Topologica l Marko v Chain s 7 1 2.5 Topologica l Pressur e , Hausdorff An d Bo x Dimension .... 8 1

3 One-Dimensiona l Dynamic s 9 5 3.1 Lorenz-typ e Map s 9 5 3.2 Continuou s an d Smoot h Map s o f the Interva l Il l 3.3 Ergodi c Propertie s an d Invarian t Measure s 13 4

4 Two-Dimensiona l Dynamic s 15 3 4.1 Henon-typ e Map s 15 3 4.2 Th e Notio n o f Hyperbolicity 15 7 4.3 Sufficien t Condition s fo r Hyperbolicit y 16 1 4.4 Poincare-Birkhof f Proble m 16 9 4.5 Homoclini c Bifurcation s 19 5 4.6 Strang e s o f Som e Map s o f the Plan e 21 0

5 System s wit h 1. 5 Degree s o f Freedo m 21 7 5.1 Smal l Periodic Perturbatio n o f Morse-Smale System s .... 21 7 5.2 Bifurcation s O f Codimension On e Subjected t o Periodic Per- turbations 22 0 5.3 Th e Melhiko v Function 22 9 5.4 Route s to Chaos . Period-Doublin g Cascad e 24 2 5.5 Critica l Saddle-Nod e Bifurcation s an d Destructio n o f Tori . 24 7

v vi CONTENTS

6 Generate d b y 3- D Vecto r Field s 25 7 6.1 Homoclini c Bifurcation s i n System s 25 7 6.2 Tw o Homoclinic Orbit s 26 6 6.3 Th e Geometri c Loren z Attracto r 27 4 6.4 Saddle-Focu s Homoclini c Bifurcation s 28 6

7 Lyapuno v Exponent s 29 5

Appendix 31 1 .1 Proo f o f the Annulu s Principl e 31 1 .2 Norma l Form fo r the Andronov-Hopf-Naimark-Sacker Bifur - cation 31 7 .3 Dissipativ e "Separatri x Map " 32 0 .4 Derivatio n o f the Zaslavsk y ma p [Z ] 32 2 .5 Concludin g Remark s o n Symboli c Dynamic s 32 5 .6 Hyperbolicit y Condition s 32 9

References 33 9

Index 35 1 Preface

There are many books related to the field o f science which is called " chaotic dynamics", an d many o f them may serve as introductory textbooks fo r stu - dents (see , fo r instance , [AAIS] , [C] , [D] , [GH] , [KH] , [MS] , [0] , [PT2] , [Ro2], [Sp] , [W ] and reference s therein) . I n order to justify (fro m ou r poin t of view) the existenc e o f the present book , le t u s observe that th e origi n o f "dynamical chaos" has three main components: (1 ) differentiable dynamic s (mainly th e par t create d b y Alekseev , Anosov , Bowen , Shil'mkov , Sinai , Smale and others); (2 ) the derivation an d study o f mathematical model s o f physical systems (Chirikov , Zaslavsk y and others in plasma physics; Loren z and other s i n meteorology ; Hayashi , Rabinovich , Chu a an d other s i n elec - trical engineering ; May , Keene r an d other s i n mathematica l biology ; an d many other specialists in different branche s o f natural sciences); (3 ) the pos- sibility o f compute r simulatio n o f "really" nonlinea r systems . I f yo u loo k carefully a t th e book s devote d t o chao s o n th e shelve s o f the bookstores , you wil l se e that the y ca n b e partitione d (mor e o r les s without difficulty ) into three groups, i n accordance wit h the point s o f view mentioned above . In ou r lectur e notes , w e have tried t o kee p the highes t possibl e leve l o f "physicaF (an d mathematical ) intuition , whil e bein g mathematicall y rig - orous. Moreover , w e have explained som e algorithm s an d formulate d som e problems fo r those who are interested i n computer stud y o f chaotic dynam - ics. Secondly, i n ou r boo k w e wer e concerne d abou t reader s wh o ar e no t familiar wit h nonlinea r dynamic s a t all . A reader (say , graduat e student ) knowing nothin g (except , o f course , som e "standard knowledge" o f func - tional analysis , ordinar y differentia l equations , etc. ) wh o goe s through al l the example s presente d i n th e book , wil l the n b e abl e t o enjo y readin g proofs o f theorems i n books suc h a s [KH] , [MS] , to understand th e mathe - matics hidde n unde r th e nicel y describe d physica l problem s i n [O] , [SUZ], will not b e afrai d o f "numerical difficultie s i n [CK] , [GHP] , etc. A t least , we hope this wil l be so. However, t o fulfil l tha t aim , w e had t o omi t a lo t o f material. Onl y a very small piece of ergodic theory has been mentioned (th e interested reade r should loo k a t [CFS] , [KH] , [Y2 ] and reference s therein) . Hamiltonia n sys - tems ar e almos t no t touche d i n ou r boo k (sam e references) . Furthemore , the attentive reader wil l see that a s soon as a problem becomes too specific , we stop considerin g it . I n brief , ou r boo k i s an introductio n an d onl y a n

vn Vlll PREFACE introduction t o the field o f dynamical chaos . Th e sam e applie s to the bib - liography. W e d o no t preten d t o mentio n al l essentia l article s an d books . We hav e omitte d man y nic e an d importan t reference s an d include d onl y those directl y relate d t o the problem s considere d i n our book . The firs t chapte r contain s th e necessar y definition s relate d t o chao s i n dissipative dynamica l systems . W e mainl y follo w th e approac h o f Taken s ([Tl], [T2] ) to define systems possessing dynamical chaos and to distinguish deterministically generate d an d rando m observables . Chapter 2 i s devote d t o th e descriptio n o f zero-dimensiona l invarian t subsets b y means o f symbolic dynamics. W e expose the mai n properties o f topological Marko v chain s an d sho w ho w t o appl y symboli c dynamic s t o the calculatio n o f the Hausdorf f dimensio n o f invariant set s resultin g fro m geometric constructions . Chapter 3 studies dynamical systems generated b y non one-to-one maps of the interval (o r o f the circle). I t begins with piecewise continuous Lorenz- type map s an d contain s the kneadin g theor y fo r suc h maps . The n w e pass over t o continuou s an d smoot h map s o f th e interva l an d presen t a shor t outline o f the well-develope d ([MS] ) theor y o f one-dimenisona l dynamica l systems. Th e res t o f the chapter contain s a summary o f ergodic propertie s of such systems . Chapters 4 through 6 are the main ones. Chapte r 4 contains the notio n of hyperbolicity and hyperbolicity criteria in specific examples. W e describe Shil'nikov's techniqu e to study th e behavior o f orbits i n a neighborhood o f a nondegenerat e homoclini c trajector y t o a fixed saddl e point , an d sho w how to appl y th e techniqu e t o solv e the Poincare-Birkhof f problem . The n we list homoclini c bifurcation s an d accompanyin g phenomena . Chapter 5 describes som e problems relate d t o system s with 1. 5 degree s of freedom. I t studie s first th e influenc e o f small periodi c perturbations t o Morse-Smale systems and systems belonging to codimension one bifurcatio n surfaces. Roughl y speaking, we show that onl y systems with homoclinic or - bits o f saddle equlibrium, bein g subjected t o a small periodic perturbation , properties o f the Melniko v function . Th e res t o f the chapte r demonstrate s the mai n route s to chaos : th e period-doubling cascad e an d the destructio n of two-dimensional tori . Chapter 6 examines the possibility o f producing dynamica l chao s in dy- namical systems generated b y vector fields on three-dimensional manifolds . It studie s system s wit h homoclini c orbit s t o saddl e an d saddle-focu s equi - libria and systems, and simila r systems. A considerable part o f the chapte r is devoted t o Lorenz-typ e attractors . The subjec t o f Chapter 7 seems to b e o f independent interest . W e give a brie f descriptio n o f Lyapunov, pointwis e an d correlatio n dimensions . W e PREFACE IX mention some notions and ideas of this chapter earlier and they are therefor e included i n the book . We want t o emphasize that th e material i s written o n differen t level s of detail: ther e are two types o f exposition o f ideas, methods and results in the book. Th e simple things are described in some detail: o n the other hand th e more cumbersom e o r specifi c thing s ar e mainl y just liste d wit h necessar y references. Fo r example, in Section 4.4 we give a complete proof o f Theorem 4.4.1 (th e Poincare-Birkhof f problem) , but i n the next section , 4.5 , we only show some difficulties i n the study o f homoclinic bifurcations. W e could not present a satisfactory descriptio n o f this subject, a s was done, fo r example , in the book [PT] . The main reason is that w e impose "boundar y conditions " of level o f detail an d simplicity , an d follo w them throughout th e book . W e do this becaus e o f the introductor y characte r o f the book . The boo k gre w mainly fro m th e note s fo r th e specia l course s which th e first autho r taugh t a t th e Georgi a Institut e o f Technology , Atlant a (1992 - 1994), and which both authors taught at the National Tsing Hua University, Hsinchu, (1996-1998) . (Som e material wa s prepared fo r a special cours e a t Northwestern University , Evansto n (1995)) . We wish to thank al l the participants o f our seminars at Nationa l Tsin g Hua University , an d the facult y an d graduat e student s fo r valuabl e discus - sions an d hel p i n preparin g th e manuscript . Th e author s ar e gratefu l t o Professor A . Katok fo r helpfu l comments , an d to Professo r M . Malkin an d Professor J . Ringland , wh o helped a great dea l a t th e fina l stag e o f prepa - ration o f the manuscript . W e thank Mrs . Ch u Ming-Hui , wh o type d ou r manuscripti, an d Mr . Tzu-Cha w We i and Mr . Yun-Hue i Tzeng who helped us make al l the figure s i n the book . The firs t autho r woul d lik e t o than k Professor s W.-W . Lin , S.-S . Lin , C.-S. Wan g an d Mrs . Tail y Hs u fo r thei r hospitalit y durin g hi s sta y i n . H e wishe s t o than k th e Nationa l Counci l o f Science , Republi c o f China, an d th e Nationa l Cente r fo r Theoretica l Science s fo r suppor t o f hi s research program . This page intentionally left blank This page intentionally left blank REFERENCES 339

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[Ya ] E. Yanagida, Branchin g o f double puls e solutions fro m sin - gle puls e solution s i n nerv e axo n equation , J . Diff . Eqs. , 6 6 (1987), 243-262 . [Z ] G.M . Zaslavsky , Th e simples t cas e o f strang e attractor , Phys. Letters , 69A , 3(1978) , 145-147 . [Zh ] A.S . Zheleznyak , A n approac h t o th e computatio n o f the topologica l entropy , in : Researc h Report s i n Physics , Nonlinear Wave s 3 , A.V . Gaponov-Grekhov , M.I . Rabi - novich, J.Engelbrecht , eds. , Springer-Verlag , Berlin , Heidel - berg, (1990) , 301-306. Index Abels formula , 21 9 Chua attractor , 15 1 absolutely continuou s closed invarian t curve , 1 5 invariant set , 13 0 codimension-one bifurcation , 15 7 absorbing region , 9 codimension-two bifurcation , 23 5 Andronov-Hopf bifurcation , 158 , Collet-Eckmann condition, 106 , 108 159 complexity, 27 , 242 Andronov-Hopf-Neimark-Sacker cone family , 245 , 246,24 8 bifurcation, 18 , 15 9 correlation dimension, 36, 213, 220, annulus principle , 16 , 18 , 225 224 Anosov diffeomorphism , 11 3 correlation integral , 22 4 Arnold tongue , 20 , 84 coupled map lattice, 214, 215, 220, attractor, 6 , 9 , 10 , 1 1 221 critical saddle-node, 178 , 180, 183 ball o f dissipation, 7 cylinder, 39 , 48 Belitskij theorem , 20 7 Belykh map, 22 0 Delay coordinate algorithm , 3 3 Benedicks-Carleson result , 14 9 destruction o f two dimensional tori, Bernoulli shif t (scheme) , 38 , 42 182 one-sided, 4 5 discritical invers e node, 21 7 two-sided, 4 5 discritical node , 21 6 bifurcation surface , 94 , 95, 97 dissipative, 7 , 173 , 235 bipermutative cellular automation , dissipative separatrix map, 17 , 161, 54 173, 23 5 Birkhoff Ergodi c Theorem , 9 9 doubling transformation, 9 6 Borel set, 9 8 Duffing equation , 8 , 17 1 Borel a-algebra, 9 8 Dyadic addin g machine , 9 1 box dimension , 59 , 63 dynamical chaos , 5 upper, 6 5 lower, 6 5 equilibrium, 1 0 Brouwer theorem , 229 , 254 ergodic, 9 7 Bykov contour, 205 , 206 established motion , 9 expansive, 6 Cantor set , 2 1 expansive map , 4 8 Cellular, 3 , 42 Cellular automata , 3 , 42 Feigenbaum attractor , 92 , 95 Chaotic repellers , 19 5 Feigenbaum universality , 9 5 Chebyshev polynomial , 21 5 niation, 26 7 Chirikov map , 15 1 fixed point , 4 , 1 1

351 INDEX 352 fractal dimension , 2 7 logistic map, 3 fundamental region , 18 1 Lorentz attractor , 25 2 Lorentz system , 8 generalized hyperboli c attractor , Lorentz-type map , 8 5 153 Lozi map, 15 1 geometric construction , 23 , 66 Lyapunov dimension , 3 6 geometric Smal e horseshoes , 112 , Lyapunov exponents , 106 , 21 3 209, 21 0 Mane Theorem , 3 0 Hadamard conditions , 109 , 15 0 Markov partition, 75 , 202 Hausdorff dimension , 64 , 66 Markov processes , 10 5 Henon map, 10 9 maximal attractor , 9 homoclinic, 14 0 measurable set , 9 8 homoclinic orbit , 84 , 155 , 19 2 Melhikov function , 16 3 homoclinic tangency, 148 , 149, 183 Minimum principle , 8 9 hyperbolic attractor , 9 0 Mobius band, 18 9 hyperbolic automorphism , 11 3 Moran construction , 6 6 hyperbolic repeller , 9 0 Morse-Smale system , 15 5 hyperbolic set , 149 , 211 multiplicative ergodi c theorem o f Oseledec, 21 6 immediate basin , 9 0 multiplier, 11 , 13 indicator function , 10 0 infinitely renormalizable , 89 , 91 neutral, 9 0 invariant, 4 Newhouse phenomena, 14 8 invariant measure , 9 7 Newhouse region , 14 9 invariant tori , 1 5 Nonlocal bifurcation , 16 0 nonwandering, 4 , 84 Jacobson Theorem , 10 8 observable, 1 Kneading invariants , 76 , 77, 87 orbit, 4 Kneading series , 7 9 oriented graph , 5 3 Kneading theory , 7 6 Kolmogorov complexity , 24 2 period-doubling bifurcation , 9 3 period-doubling cascade, 147 , 173 lambda lemma , 122 , 13 0 periodic orbit, 4 , 156 , 165 , 18 9 leading direction , 22 8 periodic perturbation, 14 , 155, 157 Lebesgue density point , 10 8 periodic point, 1 2 Lebsgue measure , 9 8 Perron-Frobenius equation, 124 , 125 limit cycle , 1 3 Perron-Frobenius operator , 12 5 Liouville's formula , 5 7 Perron-Frobenius theorem, 66, 127 local bifurcatio n o f phase space , 3 equilibrium, 19 4 Poincare-Birkhoff problem , 12 1 353 INDEX

Poincare map, 1 5 subshifts o f finite type, 5 2 Poincare recurrence theorem, 12 1 cr-algebra, 9 8 pointwise dimension , 22 3 point wise dissipative, 17 4 Taken's ratio, 3 5 Poisson bracket , 17 1 tent map , 4 0 positive semi-trajectory , 4 time-correlation, 10 5 probability measure , 9 8 topological entropy , 6 , 27 , 79 topological Markov chain, 52, 243 quadratic homoclinic tangency, 18 3 topological mixing , 6 , 5 6 quadratic tangency , 14 0 topological pressure, 5 9 quasi-hyperbolic, 15 2 topological transitive , 7 topological transitivity , 6 8 reaction-diffusion equation , 11 6 trajectory, 4 renormalizaiton, 8 8 transient process , 9 rotation number , 20 , 84 transition matrix , 53 , 243

s-critical, 22 1 Unimodal map , 8 6 S-unimodal map, 2 unstable set , 4 7 saddle-focus point , 20 5 unstable subspace , 11 5 saddle node, 18 0 saddle-node bifurcation, 146 , 158, wild hyperboli c set , 14 9 178 Schwarzian derivative , 8 9 Zaslavsky map, 16 , 19, sensitive dependenc e o n 112, 152 , 239 initial condition , 6 Zeta function , 5 7 separatrix map , 1 7 Sharkovski ordering , 10 3 Sharkovsky Theorem , 8 3 Shilnikov's technique , 12 3 sine-Gordon, 15 0 Smale attractor, 2 4 spectral decompositio n theorem , 153 stable set , 4 7 stable subspace , 11 5 standard map , 15 1 steady-state equatio n o f sine-Gordon, 15 0 steady-stable solution , 14 3 strange attractor, 1 9 strongly transitive, 7 0 structurally stable , 11 2 This page intentionally left blank Titles i n Thi s Serie s

28 Valenti n Afraimovic h an d Sze-B i Hsu , Lecture s o n Chaoti c Dynamica l Systems , 200 3 27 M . Ra m Murty , Introductio n t o p-adi c Analyti c Numbe r Theory , 200 2 26 Raymon d Chan , Yue-Kue n Kwok , Davi d Yao , an d Qian g Zhang , Editors , Applied Probability , 200 2 25 Dongga o Deng , Dare n Huang , Rong-Qin g Jia , We i Lin , an d Jia n Zhon g Wong , Editors, Wavele t Analysi s an d Applications , 200 2 24 Jan e Gilman , Willia m W . Menasco , an d Xiao-Son g Lin , Editors , Knots , Braids , and Mappin g Clas s Groups—Paper s Dedicate d t o Joa n S . Birman , 200 1 23 Cumru n Vaf a an d S.-T . Yau , Editors , Winte r Schoo l o n Mirro r Symmetry , Vecto r Bundles an d Lagrangia n Submanifolds , 200 1 22 Carlo s Berenstein , Der-Che n Chang , an d Jingzh i Tie , Laguerr e Calculu s an d It s Applications o n the Heisenber g Group , 200 1 21 Jiirge n Jost , Bosoni c Strings : A Mathematical Treatment , 200 1 20 L o Yan g an d S.-T . Yau , Editors , Firs t Internationa l Congres s o f Chines e Mathematicians, 200 1 19 So-Chi n Che n an d Mei-Ch i Shaw , Partia l Differentia l Equation s i n Severa l Comple x Variables, 200 1 18 Fangyan g Zheng , Comple x Differentia l Geometry , 200 0 17 Le i Gu o an d Stephe n S.-T . Yau , Editors , Lecture s o n Systems , Control , an d Information, 200 0 16 Rud i Weikar d an d Gilber t Weinstein , Editors , Differentia l Equation s an d Mathematical Physics , 200 0 15 Lin g Hsia o an d Zhoupin g Xin , Editors , Som e Curren t Topic s o n Nonlinea r Conservation Laws , 200 0 14 Jun-ich i Igusa , A n Introductio n t o th e Theor y o f Loca l Zet a Functions , 200 0 13 Vasilio s Alexiade s an d Georg e Siopsis , Editors , Trend s i n Mathematical Physics , 1999 12 Shen g Gong , Th e Bieberbac h Conjecture , 199 9 11 Shinich i Mochizuki , Foundation s o f p-adic Teichmulle r Theory , 199 9 10 Duon g H . Phong , Lu c Vinet , an d Shing-Tun g Yau , Editors , Mirro r Symmetr y III , 1999 9 Shing-Tun g Yau , Editor , Mirro r Symmetr y I , 199 8 8 Jiirge n Jost , Wilfri d Kendall , Umbert o Mosco , Michae l Rockner , and Karl-Theodo r Sturm , Ne w Direction s i n Dirichle t Forms , 199 8 7 D . A . Buel l an d J . T . Teitelbaum , Editors , Computationa l Perspective s o n Numbe r Theory, 199 8 6 Harol d Levine , Partia l Differentia l Equations , 199 7 5 Qi-ken g Lu , Stephe n S.-T . Yau , an d Anatol y Libgober , Editors , Singularitie s an d Complex Geometry , 199 7 4 Vyjayanth i Char i an d Iva n B . Penkov , Editors , Modula r Interfaces : Modula r Li e Algebras, Quantu m Groups , an d Li e Superalgebras , 199 7 3 Xia-X i Din g an d Tai-Pin g Liu , Editors , Nonlinea r Evolutionar y Partia l Differentia l Equations, 199 7 2.2 Willia m H . Kazez , Editor , Geometri c Topology , 199 7 2.1 Willia m H . Kazez , Editor , Geometri c Topology , 199 7 1 B . Green e an d S.-T . Yau , Editors , Mirro r Symmetr y II , 199 7