Lectures on Chaotic Dynamical System 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 Copying an d reprinting . 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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 Attractors : 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 Attractor 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 .