«ACTA»2002. h Introduction. Theory. Dissipative Systems. I. D. Chueshov. Introduction to the Theory of Infinite-Dimensional Dissipative Systems

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1 «ACTA»2002 Auhor: Tile: I D Chueshov Inroducion o he Theory of Infinie-Dimensional Dissipaive Sysems ISBN: I D Chueshov h Inroducion Theory of Infinie-Dimensional Dissipaive Sysems o hetheory Universiylecuresinconemporarymahemaics You can ORDER R his book while visiing he websie of «ACTA» Scienific Publishing House hp://wwwacacomua/en/ This book provides an exhau - sive inroducion o he scope of main ideas and mehods of he heory of infinie-dimensional dis - sipaive dynamical sysems which has been rapidly developing in re - cen years In he examples sys ems generaed by nonlinear parial differenial equaions arising in he differen problems of modern mechanics of coninua are considered The main goal of he book is o help he reader o maser he basic sraegies used in he sudy of infinie-dimensional dissipaive sysems and o qualify him/her for an independen scien - ific research in he given branch Expers in nonlinear dynamics will find many fundamenal facs in he convenien and pracical form in his book The core of he book is com - posed of he courses given by he auhor a he Deparmen of Me chanics and Mahemaics a Kharkov Universiy during a number of years This book con - ains a large number of exercises which make he main ex more complee I is sufficien o know he fundamenals of funcional analysis and ordinary differenial equaions o read he book Translaed by Consanin I Chueshov from he Russian ediion («ACTA», 1999) Translaion edied by Maryna B Khorolska

2 Chaper 6 Homoclinic Chaos in Infinie-Dimensional Sysems Conens 1 Bernoulli Shif as a Model of Chaos Exponenial Dichoomy and Difference Equaions Hyperboliciy of Invarian Ses for Differeniable Mappings Anosov s Lemma on e -rajecories Birkhoff-Smale Theorem Possibiliy of Chaos in he Problem of Nonlinear Oscillaions of a Plae On he Exisence of Transversal Homoclinic Trajecories 402 References 413

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4 In his chaper we consider some quesions on he asympoic behaviour of a discree dynamical sysem We remind (see Chaper 1) ha a discree dynamical sysem is defined as a pair ( X, S) consising of a meric space X and a coninuous mapping of X ino iself Mos asserions on he exisence and properies of aracors given in Chaper 1 remain rue for hese sysems I should be noed ha he following examples of discree dynamical sysems are he mos ineresing from he poin of view of applicaions: a) sysems generaed by monodromy operaors (period mappings) of evoluionary equaions, wih coefficiens being periodic in ime; b) sysems generaed by difference schemes of he ype -1 ( u n u n ) = = Fu ( n ), n = 0, 1, 2, ¼ in a Banach space X (see Examples 15 and 16 of Chaper 1) The main goal of his chaper is o give a sric mahemaical descripion of one of he mechanisms of a complicaed (irregular, chaoic) behaviour of rajecories We deal wih he phenomenon of he so-called homoclinic chaos This phenomenon is well-known and is described by he famous Smale heorem (see, eg, [1 3]) for finie-dimensional sysems This heorem is of general naure and can be proved for infinie-dimensional sysems Is proof given in Secion 5 is based on an infinie-dimensional varian of Anosov s lemma on e -rajecories (see Secion 4) The consideraions of his Chaper are based on he paper [4] devoed o he finie-dimensional case as well as on he resuls concerning exponenial dichoomies of infinie-dimensional sysems given in Chaper 7 of he book [5] We follow he argumens given in [6] while proving Anosov s lemma 1 Bernoulli Shif as a Model of Chaos Mahemaical simulaion of complicaed dynamical processes which ake place in real sysems requires ha he noion of a sae of chaos be formalized One of he possible approaches o he inroducion of his noion relies on a selecion of a class of explicily solvable models wih complicaed (in some sense) behaviour of rajecories Then we can associae every model of he class wih a definie ype of chaoic behaviour and use hese models as sandard ones comparing heir dynamical srucure wih a qualiaive behaviour of he dynamical sysem considered A discree dynamical sysem known as he Bernoulli shif is one of hese explicily solvable models Le m ³ 2 and le ì ü S m = íx = ( ¼, x -1, x 0, x 1, ¼) : x j Î { 1, 2, ¼, m}, j ÎZý, î þ ie S m1 is a se of wo-sided infinie sequences he elemens of which are he inegers, 2, ¼, m Le us equip he se Sm wih a meric

5 366 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 x dx (, y) 2 - i i - y i = å (11) 1 + x i - y i = - i Here x = { x i : i ÎZ} and y = { y i : i ÎZ} are elemens of S m Oher mehods of inroducion of a meric in S m are given in Example 117 and Exercise 115 Exercise 11 Show ha he funcion americ d( x, y) saisfies all he axioms of Exercise 12 Le x = { x i } and y = { y i } be elemens of he se S m Assume ha x i = y i for i N and for some ineger N Prove ha d( x, y) 2 -N + 1 Exercise 13 Assume ha equaion d( x, y) < 2 -N holds for x, y Î S m, where N is a naural number Show ha x i = y i for all i N-1 (Hin: d( x, y) ³ 2 - i -1 if x i ¹ y i ) Exercise 14 Le x Î S m and le U N ( x) = { y Î S m : x i = y i for i N} (12) Prove ha for any 0 < e < 1 he relaion U N( e) + 2 ( x) Ì { y : dx (, y) < e} Ì U N( e) -1 ( x) holds, where N() e is an ineger wih he propery ln1 e N( e) < N( e) + 1 ln2 wih meric (11) is a compac me- Exercise 15 Show ha he space S m ric space In he space S m we define a mapping S which shifs every sequence one symbol lef, ie [ Sx] i = x i + 1, i Î Z, x = { x i } Î S m Evidenly, S is inverible and he relaions d( Sx, Sy) 2 dx (, y), d( S -1x, S-1 y) 2 dx (, y) hold for all x, y Î S m Therefore, he mapping S is a homeomorphism The discree dynamical sysem ( S m, S) is called he Bernoulli shif of he space of sequences of m symbols Le us sudy he dynamical properies of he sysem ( S m, S) Exercise 16 Prove ha ( S m, S) has m fixed poins exacly Wha srucure do hey have?

6 Bernoulli Shif as a Model of Chaos 367 We call an arbirary ordered collecion a = ( a 1, ¼, a N ) wih a j Î { 1, ¼, m} a segmen (of he lengh N) Each elemen x Î S m can be considered as an ordered infinie family of finie segmens while he elemens of he se S m can be consruced from segmens In paricular, using he segmen a = ( a 1, ¼, a N ) we can consruc a periodic elemen a Î S m by he formula a Nk+ j = a j, j Î { 1, ¼, N}, k Î Z (13) Exercise 17 Le a = ( a 1, ¼, a N ) be a segmen of he lengh N and le a Î S m be an elemen defined by (13) Prove ha a is a periodic poin of he period N of he dynamical sysem ( S m, S), ie S N a = a Exercise 18 Prove ha for any naural of he minimal period equal o N N here exiss a periodic poin Exercise 19 Prove ha he se of all periodic poins is dense in S m, ie for every x ÎS m and e > 0 here exiss a periodic poin a wih he propery d( x, a) < e (Hin: use he resul of Exercise 14) Exercise 110 Prove ha he se of nonperiodic poins is no counable M 1 M 2 Exercise 111 Le a = ( ¼, a, a, a, ¼) and b = ( ¼, b, b, b, ¼) be fixed poins of he sysem ( S m, S) Le C = { c i } be an elemen of S m such ha c i = a for i -M 1 and c i = b for i ³ M 2, where and are naural numbers Prove ha lim S n c = a, lim S n c = b (14) n - n Assume ha an elemen c Î S m possesses propery (14) wih c ¹ a and c ¹ b If a ¹ b, hen he se g a, b = { S n c : n Î Z} is called a heeroclinic rajecory ha connecs he fixed poins a and b If a = b, hen g a = g a, a is called a homoclinic rajecory of he poin a The elemens of a heeroclinic (homoclinic, respecively) rajecory are called heeroclinic (homoclinic, respecively) poins Exercise 112 Prove ha for any pair of fixed poins here exiss an infinie number of heeroclinic rajecories connecing hem whereas he corresponding se of heeroclinic poins is dense in S m Exercise 113 and Le g 1 = { S n a : n ÎZ} = { S n a : n = 0, 1, ¼, N 1-1} g 2 = { S n b : n Î Z} = { S n b : n = 0, 1, ¼, N 2-1}

7 368 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 be cycles (periodic rajecories) Prove ha here exiss a heeroclinic rajecory g 1, 2 = { S n c : n Î Z} ha connecs he cycles g 1 and g 2, ie such ha dis( S n c, g 1 ) º inf d( S n c, x) 0, n - x Î g 1 and dis( S n c, g 2 ) = inf d( S n c, x) 0, n + x Î g 2 For every N here exiss only a finie number of segmens of he lengh N Therefore, he se L of all segmens is counable, ie we can assume ha L = { a k : k = 1, 2, ¼}, herewih he lengh of he segmen a k + 1 is no less han he lengh of a k Le us consruc an elemen b = { b i : i ÎZ} from S m aking b i = 1 for i 0 and sequenially puing all he segmens a k o he righ of he zeroh posiion As a resul, we obain an elemen of he form b = ( ¼, 1, 1, 1, a 1, a 2, a 3, ¼), a j Î L (15) Exercise 114 Prove ha a posiive semirajecory g + = { S n b, n ³ 0} wih b having he form (15) is dense in S m, ie for every x ÎS m and e > 0 here exiss n = n( x, e) such ha d( x, S n b) < e Exercise 115 Prove ha he semirajecory g + consruced in Exercise 114 reurns o an e -viciniy of every poin x ÎS m infinie number of imes (Hin: see Exercises 14 and 19) Exercise 116 Consruc a negaive semirajecory g = { S n c : n 0} which is dense in S m Thus, summing up he resuls of he exercises given above, we obain he following asserion Theorem 11 The dynamical sysem ( S m, S) of he Bernoulli shif of sequences of m symbols possesses he properies: 1) here exiss a finie number of fixed poins; 2) here exis periodic orbis of any minimal period and he se of hese orbis is dense in he phase space S m ; 3) he se of nonperiodic poins is uncounable; 4) heeroclinic and homoclinic poins are dense in he phase space; 5) here exis everywhere dense rajecories All hese properies clearly imply he exraordinariy and complexiy of he dynamics in he sysem ( S m, S) They also give a moivaion for he following definiions

8 Exponenial Dichoomy and Difference Equaions 369 Le ( X, f) be a discree dynamical sysem The dynamics of he sysem ( X, f) is called chaoic if here exiss a naural number k such ha he mapping f k is opologically conjugae o he Bernoulli shif for some m, ie here exiss a homeomorphism h: X S m such ha h( f k ( x) ) = Shx ( ( )) for all x Î X We also say ha chaoic dynamics is observed in he sysem ( X, f) if here exis a number k and a se Y in X invarian wih respec o f k ( f k Y Ì Y) such ha he resricion of f k o Y is opologically conjugae o he Bernoulli shif I urns ou ha if a dynamical sysem ( X, f) has a fixed poin and a corresponding homoclinic rajecory, hen chaoic dynamics can be observed in his sysem under some addiional condiions (his asserion is he core of he Smale heorem) Therefore, we ofen speak abou homoclinic chaos in his siuaion I should also be noed ha he approach presened here is jus one of he possible mehods used o describe chaoic behaviour (for example, oher approaches can be found in [1] as well as in book [7], he laer conains a survey of mehods used o sudy he dynamics of complicaed sysems and processes) 2 Exponenial Dichoomy and Difference Equaions This is an auxiliary secion Nonauonomous linear difference equaions of he form x n + 1 = A n x n + h n, n Î Z, (21) in a Banach space X are considered here We assume ha { A n } is a family of linear bounded operaors in X, h n is a sequence of vecors from X Some resuls boh on he dichoomy (spliing) of soluions o homogeneous ( h n º 0) equaion (21) and on he exisence and properies of bounded soluions o nonhomogeneous equaion are given here We mosly follow he argumens given in book [5] as well as in paper [4] devoed o he finie-dimensional case Thus, le { A n : n Î Z} be a sequence of linear bounded operaors in a Banach space X Le us consider a homogeneous difference equaion x n + 1 = A n x n, n Î J, (22) where J is an inerval in Z, ie a se of inegers of he form J = { n Î Z : m 1 < n < m 2 }, where m 1 and m 2 are given numbers, we allow he cases m 1 = - and m 2 = + Evidenly, any soluion { x n : n Î J} o difference equaion (22) possesses he propery x m = ( m, n)x n, m ³ n, m, n Î J,

9 370 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 where ( m, n) = A m -1 ¼ A n for m > n and ( m, m) = I The mapping ( m, n) is called an evoluionary operaor of problem (22) Exercise 21 Prove ha for all m ³ n ³ k we have ( m, k) = ( m, n) ( n, k) Exercise 22 Le { P n : n Î J} be a family of projecors (ie P2 n = P n ) in X such ha P n + 1 A n = A n P n Show ha P m ( m, n) = ( m, n)p n, m ³ n, m, n Î J, ie he evoluionary operaor ( m, n) maps P n X ino P m X Exercise 23 Prove ha soluions { x n } o nonhomogeneous difference equaion (21) possess he propery m - 1 x m = ( m, n)x n + å ( m, k+ 1)h k, m > n k= n Le us give he following definiion A family of linear bounded operaors { A n } is said o possess an exponenial dichoomy over an inerval J wih consans K > 0 and 0 < q < 1 if here exiss a family of projecors { P n : n Î J} such ha a) P n + 1 A n = A n P n, n, n+ 1 Î J ; b) ( m, n)p n Kq m- n, m ³ n, m, n Î J ; c) for n ³ m he evoluionary operaor ( n, m) is a one-o-one mapping of he subspace P m )X ono ( 1 - P n )X and he following esimae holds: ( n, m) P n ) Kq n- m, m n, m, n Î J If hese condiions are fulfilled, hen i is also said ha difference equaion (22) admis an exponenial dichoomy over J I should be noed ha he cases J = Z and J = Z ± are he mos ineresing for furher consideraions, where Z + ( Z ) is he se of all nonnegaive (nonposiive) inegers The simples case when difference equaion (22) admis an exponenial dichoomy is described in he following example Example 21 (auonomous case) Assume ha equaion (22) is auonomous, ie A n º A for all n, and he specrum s ( A) does no inersec he uni circumference { z Î C : z =1} Linear operaors possessing his propery are ofen called hyperbolic (wih respec o he fixed poin x = 0 ) I is well-known (see, eg, [8]) ha in his case here exiss a projecor P wih he properies:

10 Exponenial Dichoomy and Difference Equaions 371 a) AP = PA, ie he subspaces PX and P)X are invarian wih respec o A ; b) he specrum s ( A PX ) of he resricion of he operaor A o PX lies sricly inside of he uni disc; c) he specrum s A P)X of he resricion of o he subspace P)X lies ouside he uni disc Exercise 24 Le C be a linear bounded operaor in a Banach space X and le r º max { z : z Î s( C) } be is specral radius Show ha for any q > r here exiss a consan M q ³ 1 such ha C n M q q n, n = 0, 1, 2, ¼ (Hin: use he formula r C n 1/ n = lim he proof of which can be n found in [9], for example) Applying he resul of Exercise 24 o he resricion of he operaor A o PX, we obain ha here exis K > 0 and 0 < q < 1 such ha A n P Kq n, n ³ 0 (23) I is also eviden ha he resricion of he operaor A o P)X is inverible and he specrum of he inverse operaor lies inside he uni disc Therefore, A -n P) Kq n, n ³ 0, (24) where he consans K > 0 and 0 < q < 1 can be chosen he same as in (23) The evoluionary operaor ( m, n) of he difference equaion x n + 1 = Ax n has he form ( m, n) = A m- n, m ³ n Therefore, he equaliy AP = PA and esimaes (23) and (24) imply ha he equaion x n + 1 = Ax n admis an exponenial dichoomy over Z, provided he specrum of he operaor A does no inersec he uni circumference Exercise 25 Assume ha for he operaor A here exiss a projecor P such ha AP = PA and esimaes (23) and (24) hold wih 0 < q < 1 Show ha he specrum of he operaor A does no inersec he uni circumference, ie A is hyperbolic Thus, he hyperboliciy of he linear operaor A is equivalen o he exponenial dichoomy over Z of he difference equaion x n + 1 = Ax n wih he projecors P n independen of n Therefore, he dichoomy propery of difference equaion (22) should be considered as an exension of he noion of hyperboliciy o he nonauonomous case The meaning of his noion is explained in he following wo exercises Exercise 26 Le A be a hyperbolic operaor Show ha he space X can be decomposed ino a direc sum of sable X s and unsable X u subspaces, ie X = X s + X u herewih

11 372 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 A n x Kq n x, x Î X s, n ³ 0, A n x ³ K -1 q -n x, x Î X u, n ³ 0, wih some consans K > 0 and 0 < q < 1 Exercise 27 Le X = R 2 be a plane and le A be an operaor defined by he formula Ax ( 1, x 2 ) = ( 2 x 1 + x 2 ; x 1 + x 2 ), x = ( x 1 ; x 2 ) Î R 2 Show ha he operaor A is hyperbolic Evaluae and display graphically sable X s and unsable X u subspaces on he plane Display graphically he rajecory { A n x : n Î Z} of some poin x ha lies neiher in X s, nor in X u The nex asserion (is proof can be found in he book [5]) plays an imporan role in he sudy of exisence condiions of exponenial dichoomy of a family of operaors { A n : n Î Z} Theorem 21 Le { A n : n Î Z} be a sequence of linear bounded operaors in a Ba- nach space X Then he following asserions are equivalen: (i) he sequence { A n : n Î Z} possesses an exponenial dichoomy over Z, (ii) for any bounded sequence { h n : n Î Z} from X here exiss a unique bounded soluion { x n : n Î Z} o he nonhomogeneous difference equaion x n + 1 = A n x n + h n, n Î Z (25) In he case when he sequence { A n } possesses an exponenial dichoomy, soluions o difference equaion (25) can be consruced using he Green funcion: Exercise 28 G( n, m) ì ï = í ï î ( n, m)p m, n ³ m, -[ ( m, n) ] -1 ( 1 - P m ), n < m Prove ha Gn (, m) Kq n- m Exercise 29 Prove ha for any bounded sequence { h n : n Î Z} from X a soluion o equaion (25) has he form x n = å G( n, m + 1)h m, n Î Z m Î Z

12 Exponenial Dichoomy and Difference Equaions 373 Moreover, he following esimae is valid: sup x n K q sup n 1- q n h n + The properies of he Green funcion enable us o prove he following asserion on he uniqueness of he family of projecors { P n } Lemma 21 Le a sequence { A n } possess an exponenial dichoomy over Z Then he projecors { P n : n Î Z} are uniquely defined Proof Assume ha here exis wo collecions of projecors { P n } and { Q n } for which he sequence { A n } possesses an exponenial dichoomy Le G P ( n, m) and G Q ( n, m) be Green funcions consruced wih he help of hese collecions Then Theorem 21 enables us o sae (see Exercise 29) ha å G P n, m+ 1 = m å G Q ( n, m+ 1)h m m Î Z m Î Z for all n Î Z and for any bounded sequence { h n } Ì X Assuming ha h m = 0 for m ¹ k-1 and h m = h for m = k-1, we find ha G P ( n, k)h = G Q ( n, k)h, h Î X, n, k Î Z, n ³ k This equaliy wih n = k gives us ha P n h = Q n h Thus, he lemma is proved In paricular, Theorem 21 implies ha in order o prove he exisence of an exponenial dichoomy i is sufficien o make sure ha equaion (25) is uniquely solvable for any bounded righ-hand side I is convenien o consider his difference equaion in he space l X º l ( Z, X) of sequences x = { x n : n ÎZ} of elemens of X for which he norm ì ü x (26) l = { x n } = sup l í x n : n Î Zý î þ is finie Assume ha he condiion ì ü sup í A n : n Î Zý < (27) î þ is valid Then for any x = { x n } Îl X he sequence { y n = x n - A n -1 x n -1 } lies in l X Consequenly, equaion ( L x) n = x n - A n -1 x n -1, x = { x n } Î l X (28) defines a linear bounded operaor acing in he space l X = l ( Z, X) Therewih asserion (ii) of Theorem 21 is equivalen o he asserion on he inveribiliy of he operaor L given by equaion (28)

13 374 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 The asserion given below provides a sufficien condiion of inveribiliy of he operaor L Due o Theorem 21 his condiion guaranees he exisence of an exponenial dichoomy for he corresponding difference equaion This asserion will be used in Secion 4 in he proof of Anosov s lemma I is a slighly weakened varian of a lemma proved in [6] Theorem 22 Assume ha a sequence of operaors { A n : n Î Z} saisfies condiion (27) Le here exis a family of projecors { Q n : n Î Z} such ha Q n K, 1 - Q n K, (29) Q n + 1 A n ( 1 - Q n ) d, Q n + 1 ) A n Q n d, (210) for all n Î Z We also assume ha he operaor Q n + 1 ) A n is inverible as a mapping from Q n )X ino Q n + 1 )X and he esimaes A n Q n l, [ Q n + 1 )A n ] -1 ( 1 - Q (211) n + ) l 1 are valid for every n Î Z If K l 1 --, d 1 --, (212) 8 8 hen he operaor L acing in l X according o formula (28) is inverible and L -1 2 K + 1 Proof Le us firs prove he injeciviy of he mapping L Assume ha here exiss a nonzero elemen x = { x n } Îl X such ha Lx = 0, ie x n A n -1 x for all n -1 Î Z Le us prove ha he sequence { x n } possesses he propery Q n )x n Q n x n (213) for all n Î Z Indeed, le here exis m Î Z such ha Q m )x m > Q m x m (214) I is eviden ha his equaion is only possible when Q m )x m > 0 Le us consider he value N m + 1 º Q m + 1 )x m Q m + 1 x m + 1 = = Q m + 1 )A m x m - Q m + 1 A m x m (215) I is clear ha Q n + 1 )A n x n ³ Q n + 1 )A n Q n )x n - Q n + 1 )A n Q n x n Since [ Q n + 1 )A n ] ( 1 - Q n + ) A 1 n Q n ) = Q n ),

14 Exponenial Dichoomy and Difference Equaions 375 i follows from (211) ha Q n )x l Q n + 1 ) A n Q n )x for every x Î X and for all n Î Z Therefore, we use esimaes (210) o find ha Q n + 1 )A n x n ³ l Q n )x n - d x n (216) Then i is eviden ha Q n + 1 A n x n Q n + 1 A n Q n x n + Q n + 1 A n Q n )x n æ Q è n + 1 A n Q n + Q n + 1 A n Q n ) ö ø x n Therefore, esimaes (29) (211) imply ha Q n + 1 A n x n ( K l + d) x n, n Î Z (217) Thus, equaions (215) (217) lead us o he esimae N m + 1 ³ l Q m )x m - ( 2 d + K l) x m I follows from (214) ha x m Q m x m + Q m )x m < 2 Q m )x m Therefore, N m + 1 > ( l - 2K l - 4d) Q m )x m Hence, if condiions (212) hold, hen Q m + 1 )x m Q m + 1 x m + 1 > 7 ( 1 - Q )x m m > 0 (218) When proving (218) we use he fac ha l ³ 8K ³ 8 Q n ³ 8 Thus, equaion (218) follows from (214), ie N m > 0 implies N m + 1 > 0 Hence, Q n )x n > Q n x n for all n ³ m Moreover, (218) gives us ha K x n ³ Q n )x n ³ 7 n-m Q m )x m, n ³ m Therefore, x n + as n + This conradics he assumpion x = { x n } Îl X Thus, for all n Î Z esimae (213) is valid In paricular i leads us o he inequaliy x n Q n )x n + Q n x n 2 Q n x n (219) Therefore, i follows from (217) ha Q n + 1 x n + 1 = Q n + 1 A n x n 2 ( K l + d) Q n x n for all n Î Z We use condiions (212) o find ha Q n + 1 x n Q, 2 n x n n Î Z (220)

15 376 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 If x = { x n } ¹ 0, hen inequaliy (219) gives us ha here exiss m Î Z such ha Q m x m ¹ 0 Therefore, i follows from (220) ha for all n m We end n - o obain ha Q n x n + which is impossible due o (29) and he boundedness of he sequence { x n } Therefore, here does no exis a nonzero x Î l X such ha L x = 0 Thus, he mapping L is injecive Le us now prove he surjeciviy of L Le us consider an operaor R in he space l X acing according o he formula ( R y) n = Q n y n - B n Q n + 1 )y n + 1, y = { y n } Î l X, where he operaor B n = [ Q n + 1 )A n ] -1 acs from Q n + 1 )X ino Q n )X and is inverse o Q n + 1 )A n I follows from (29) and (211) ha ( 1 - Qn )X Ry, (221) l ( K + l) y l y Î l X I is eviden ha Q n x n ³ 2 m- n Q m x m > 0 ( LRy) n - y n = - Q n ) y n - B n Q n 1 )y n Since we have ha - A n -1 Q n -1 y n -1 A n -1 B n -1 Q n )y n Q n ) A n -1 B n -1 Q n ) = 1- Q n, ( LRy) n - y n = -B n Q n 1 )y n 1 - A n Q n A n -1 Q n -1 )B n -1 Q n )y n Consequenly, + Q n A n -1 Q n -1 ) B n -1 Q n ) y n Therefore, inequaliies (210), (211), and (212) give us ha 1 LRy y - y, l l ( 2 + d) y l -- y 2 l ie LR Tha means ha he operaor LR is inverible and ( LR) -1 ( 1 - LR - 1 ) -1 2 (222) Le h = { h n } be an arbirary elemen of l X Then i is eviden ha he elemen y = = R( LR) -1 h is a soluion o equaion Ly = h Moreover, i follows from (221) and (222) ha y l 2 ( K + l) h l Hence, L is surjecive and L -1 2 K + 1 Theorem 22 is proved + - Q n -1 y n -1 ( LRy) n - y n B n Q n 1 ) y n 1 + A n Q n -1 y n -1

16 Hyperboliciy of Invarian Ses for Differeniable Mappings Hyperboliciy of Invarian Ses for Differeniable Mappings Le us remind he definiion of he differeniable mapping Le X and Y be Banach spaces and le U be an open se in X The mapping f from U ino Y is called (Freché) differeniable a he poin x Î U if here exiss a linear bounded operaor Df( x) from X ino Y such ha 1 lim f( x + v) - f( x) - Df( x)v = 0 v 0 v If he mapping f is differeniable a every poin x ÎU, hen he mapping Df: x Df( x) acs from U ino he Banach space L( X, Y) of all linear bounded operaors from X ino Y If Df : U L( X, Y) is coninuous, hen he mapping f is said o be coninuously differeniable (or C 1 -mapping) on U The noion of he derivaive of any order can also be inroduced by means of inducion For example, D 2 f( x) is he Freché derivaive of he mapping Df: U L( X, Y) Exercise 31 Le g and f be coninuously differeniable mappings from U Ì X ino Y and from W Ì Y ino Z, respecively Moreover, le U and W be open ses such ha g( U) Ì W Prove ha ( f g) ( x) = = f( g( x) ) is a C 1 -mapping on U and obain a chain rule for he differeniaion of a composed funcion D( f g) ( x) = Df( g( x) )Dg( x), x Î U Exercise 32 Le f be a coninuously differeniable mapping from X ino X and le f n be he n-h degree of he mapping f, ie f n ( x) = = f( f n ( x) ), n ³ 1, f 1 ( x) º f( x) Prove ha f n is a C 1 -mapping on X and ( Df n )( x) = Df( f n ( x) ) Df( f n ( x) ) ¼ Df( x) (31) Now we give he definiion of a hyperbolic se Assume ha f is a coninuously differeniable mapping from a Banach space X ino iself and L is a subse in X which is invarian wih respec o f ( f( L) Ì L) The se L is called hyperbolic (wih respec o f ) if here exiss a collecion of projecors { P( x): x Î L} such ha a) Px ( ) coninuously depends on x ÎL wih respec o he operaor norm; b) for every x Î L Df( x) P( x) = P( f( x) ) Df( x) ; (32) c) he mappings Df( x) are inverible for every x Î L as linear operaors from Px ( ))X ino ( I- P( f( x) ))X ;

17 378 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 d) for every x Î L he following equaions hold: ( Df n )( x)p( x) Kq n, n ³ 0, (33) [( Df n )( x) ] -1 P( f n ( x) )) Kq n, n ³ 0, (34) wih he consans K > 0 and 0 < q < 1 independen of x Î L Here f n is he n-h degree of he mapping f ( f n ( x) = f( f n -1 ( x) ) for n ³ 1 and f 1 ( x) º f( x) ) I should be noed ha properies (b) and (c) as well as formula (31) enable us o sae ha ( Df n )( x) maps Px ( ))X ino P( f n ( x) )) and is an inverible operaor Therefore, he value in he lef-hand side of inequaliy (34) exiss Exercise 33 Le L = { x 0 }, where x 0 is a fixed poin of a C 1 -mapping f, ie f( x 0 ) = x 0 Then for he se L o be hyperbolic i is necessary and sufficien ha he specrum of he linear operaor Df( x 0 ) does no inersec he uni circumference (Hin: see Example 21) Le L be an invarian hyperbolic se of a C 1 -mapping f and le g = { x n : n Î Z} be a complee rajecory (in L ) for f, ie g = { x n } is a sequence of poins from L such ha f( x n ) = x n + 1 for all n Î Z Le us consider a difference equaion obained as a resul of linearizaion of he mapping f along g : u n + 1 = Df( x n )u n, n Î Z (35) Exercise 34 Prove ha he evoluionary operaor ( m, n) of difference equaion (35) has he form ( m, n) = ( Df m- n )( x n ), m > n, m, n Î Z Exercise 35 Prove ha difference equaion (35) admis an exponenial dichoomy over Z wih (i) he consans K and q given by equaions (33) and (34) and (ii) he projecors P n = Px ( n ) involved in he definiion of he hyperboliciy I should be noed ha propery (a) of uniform coninuiy implies ha he projecors P( x) are similar o one anoher, provided he values of x are close enough The proof of his fac is based on he following asserion Lemma 31 Le P and Q be projecors in a Banach space X Assume ha P K, 1- P K, P- Q < (36) 2K for some consan K ³ 1 Then he operaor J = PQ + P) Q) (37)

18 Hyperboliciy of Invarian Ses for Differeniable Mappings 379 possesses he propery PJ = JQ and is inverible, herewih J -1 2K P- Q ) -1 (38) Proof Since P 2 + P) = 1, we have J -1 = J- P 2 - P) = P( Q- P) + P) ( P- Q) I follows from (36) ha J K P- Q < 1 Hence, he operaor J can be defined as he following absoluely convergen series J = å J) n = 0 This implies esimae (38) The permuabiliy propery PJ = JQ is eviden Lemma 31 is proved Exercise 36 Le L be a conneced compac se and le { Px ( ): x Î L} be a family of projecors for which condiion (a) of he hyperboliciy definiion holds Then all operaors Px ( ) are similar o one anoher, ie for any x, y Î L here exiss an inverible operaor J = J x, y such ha P( x) = JP( y) J -1 The following asserion conains a descripion of a siuaion when he hyperboliciy of he invarian se is equivalen o he exisence of an exponenial dichoomy for difference equaion (35) (cf Exercise 35) Theorem 31 Le f( x) be a coninuously differeniable mapping of he space X ino iself Le x 0 be a hyperbolic fixed poin of f ( f( x 0 ) = x 0 ) and le { y n : n ÎZ} be a homoclinic rajecory (no equal o x 0 ) of he mapping f, ie f( y n ) = y n + 1, n Î Z, y n x 0, n ± (39) Then he se L = { x 0 } È { y n : n Î Z} is hyperbolic if and only if he diffe- rence equaion u n + 1 = Df( y n )u n, n Î Z, (310) possesses an exponenial dichoomy over Z Proof If L is hyperbolic, hen (see Exercise 35) equaion (310) possesses an exponenial dichoomy over Z Le us prove he converse asserion Assume ha equaion (310) possesses an exponenial dichoomy over Z wih projecors { P n : n ÎZ}

19 380 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 and consans K and q Le us denoe he specral projecor of he operaor Df( x 0 ) corresponding o he par of he specrum inside he uni disc by P Wihou loss of generaliy we can assume ha [ Df( x 0 )] n P Kq n, n ³ 0, [ Df( x 0 )] -n P) Kq -n, n ³ 0 Thus, for every x Î L he projecor Px ( ) is defined: Px ( 0 ) = P, P( y k ) = P k The srucure of he evoluionary operaor of difference equaion (310) (see Exercise 34) enables us o verify properies (b) (d) of he definiion of a hyperbolic se In order o prove propery (a) i is sufficien o verify ha P k - P 0 as k ± (311) Since L is a compac se, hen M = sup { Df( x) : x Î L} < (312) Le us consider he following difference equaions v n + 1 = Df( x 0 )v n, n Î Z, (313) and ( k) w n + 1 Df y ( k ) = n+ k, n Î Z, (314) where k is an ineger I is eviden ha equaion (314) admis an exponenial dichoomy over Z wih consans K ( k) and q and projecors P n P Le n+ k (, m ) and G k) ( n, m) be he Green funcions (see Secion 2) of difference equaions (313) and (314) We consider he sequence x n = G( n, 0)z - G k) ( n, 0)z, z Î X Since (see Exercise 28) Gn (, 0) Kq n, G ( k) ( n, 0) Kq n, (315) we have ha he sequence { x n } is bounded Moreover, i is easy o prove (see Exercise 29) ha { x n } is a soluion o he difference equaion x n Df( x 0 )x n = h n º [ Df( x 0 ) - Df( y n+ k )]G ( k) ( n, 0)z I follows from (312) and (315) ha he sequence { h n } is bounded Therefore, (see Exercise 29), Gn (, 0)z - G ( k) ( n, 0)z º x n = å G( n, m+ 1)h m m Î Z If we ake n = 0 in his formula, hen from he definiion of he Green funcion we obain ha ( P- P k )z = å G( 0, m + 1) ( Df( x 0 )- Df( y m+ k )) G ( k) ( m, 0)z m Î Z

20 Hyperboliciy of Invarian Anosov s Ses for Lemma Differeniable on e -rajecories Mappings 381 Therefore, equaion (315) implies ha å ( P- P k )z K 2 Df( x 0 ) - Df( y m+ k ) q m + m+ 1 z m Î Z Consequenly, å P- P k K 2 Df( x 0 ) - Df( y m+ k ) q 2 m -1 m Î Z K 2 ì Df( x 0 )- Df( y m+ k ) å q 2 m 2 M q 2 m ü í max + å ý, î m N m N m > N þ where N is an arbirary naural number Upon simple calculaions we find ha P- P k 2 K 2 q æ 1 q 2 max ( - ) ( x )- Df( y 0 m+ k ) + 2 Mq 2 N 2 ö è m N ø for every N ³ 1 I follows ha lim 4 K P- P M k , k ± 1 q 2 q 2 N 1 - N = 1, 2, ¼ We assume ha N + o obain ha lim P- P k k ± 0 This implies equaion (311) Therefore, Theorem 31 is proved I should be noed ha in he case when he se L = { x 0 } È { y n : n Î Z} from Theorem 31 is hyperbolic he elemens y n of he homoclinic rajecory g = { y n : n ÎZ} are called ransversal homoclinic poins The poin is ha in some cases (see, eg, [4]) i can be proved ha he hyperboliciy of L is equivalen o he ransversaliy propery a every poin y n of he sable W s ( x 0 ) and unsable W u ( x 0 ) manifolds of a fixed poin x 0 (roughly speaking, ransversaliy means ha he surfaces W s ( x 0 ) and W u ( x 0 ) inersec a he poin y n a a nonzero angle) In his case he rajecory g is ofen called a ransversal homoclinic rajecory 4 Anosov s Lemma on e -rajecories Le f be a C 1 -mapping of a Banach space X ino iself A sequence { y n : n Î Z} in X is called a d -pseudorajecory (or d-pseudoorbi) of he mapping f if for all n Î Z he equaion - ( ) d y n + 1 f y n

21 382 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 is valid A sequence { x n : n Î Z} is called an e -rajecory of he mapping f corresponding o a d -pseudorajecory { y n : n Î Z} if (a) f( x n ) = x n + 1 for any n Î Z ; (b) x n -y n e for all n Î Z I should be noed ha condiion (a) means ha { x n } is an orbi (complee rajecory) of he mapping f Moreover, if a pair of C 1 -mappings f and g is given, hen he noion of he e -rajecory of he mapping g corresponding o a d -pseudoorbi of he mapping f can be inroduced The following asserion is he main resul of his secion Theorem 41 Le f be a C 1 -mapping of a Banach space X ino iself and le L be a hyperbolic invarian ( f ( L) Ì L) se Assume ha here exiss a D -vicini- y O of he se L such ha f( x) and Df( x) are bounded and uniformly coninuous on he closure O of he se O Then here exiss e 0 > 0 posses- sing he propery ha for every 0 < e e 0 here exiss d = d( e) > 0 such ha any d -pseudoorbi { y n : n Î Z} lying in L has a unique e -rajecory { x n : n Î Z} corresponding o { y n } As he following heorem shows, he propery of he mapping f o have an e -rajecory is rough, ie his propery also remains rue for mappings ha are close o f Theorem 42 Assume ha he hypoheses of Theorem 41 hold for he mapping f Le W h ( f ) be a se of coninuously differeniable mappings g of he space X ino iself such ha he following esimaes hold on he closure O of he D -viciniy O of he se L : f( x) -gx ( ) < h, Df( x) -Dg( x) < h (41) Then e 0 > 0 can be chosen o possess he propery ha for every e Î ( 0, e 0 ] here exis d = d( e) > 0 and h = h( e) > 0 such ha for any d -pseudora- jecory { y n : n Î Z} (lying in L ) of he mapping f and for any g Î W h ( f ) here exiss a unique rajecory { x n : n Î Z} of he mapping g wih he pro- pery y n -x n e for all n Î Z I is clear ha Theorem 41 is a corollary of Theorem 42 he proof of which is based on he lemmaa below

22 Hyperboliciy of Invarian Anosov s Ses for Lemma Differeniable on e -rajecories Mappings 383 Lemma 41 Le U be an open se in a Banach space X and le : U X be a coninuously differeniable mapping Assume ha for some poin y Î U here exis an operaor [ D ( y) ] and a number e 0 > 0 such ha D ( x) - D ( y) æ2 [ D ( y) ] ö è ø (42) for all x wih he propery x-y e 0 Assume ha for some e Î( 0, e 0 ] he inequaliy ( y) q e æ2 [ D ( y) ] ö è ø (43) is valid wih 0 < q < 1 Then for any C 1 -mapping G : U X such ha G ( x) - ( x) e q) æ2 [ D ( y) ] ö è ø (44) and DG( x) - D ( x) 1 -- æ2 [ D ( y) ] ö 2 è ø (45) for x-y e 0, he equaion G ( x) = 0 has a unique soluion x wih he propery x-y e Proof Le G = ( 2 [ D ( y) ] ) and le h( x) = ( x) - ( y) - D ( y) ( x- y) For x 1, x 2 Î B e0 º { z : z-y e 0 } we have ha h( x 1 )- h( x 2 ) = ( x 1 ) - ( x 2 ) - D ( y) ( x 1 - x 2 ) = 1 ò = 0 Since ( D ( x 1 + x( x 2 - x 1 )) - D ( y) )( x 1 - x 2 ) dx 1 ò h( x 1 ) - h( x 2 ) D ( x 1 + x ( x 2 - x 1 )) -D ( y) dx x 1 - x 2 0 i follows from (42) ha h( x 1 ) -h( x 2 ) G x 1 - x 2, (46) for all x 1 and x 2 from B e0 Now we rewrie he equaion G( x) = 0 in he form x = T( x) º y- [ D ( y) ] -1 ( G( x) - ( x) + ( y) + h( x) )

23 384 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 Le us show ha he mapping T has a unique fixed poin in he ball B e = { x : x-y e} I is eviden ha Tx ( ) - y [ D ( y) ] -1 æ G( x) - ( x) + ( y) + h( x) ö è ø for any x ÎB e Since h( y) = 0, we obain from (46) ha h( x) = h( x) -h( y) G x-y Ge Therefore, esimaes (43) and (44) imply ha Tx ( )-y e for x Î B e, ie T maps he ball B e ino iself This mapping is conracive in B e Indeed, Tx ( 1 )- Tx ( 2 ) æ H ( x, 2G 1, x 2 ) + h( x 1 )- h( x 2 ) ö è ø where H ( x 1, x 2 ) º G ( x 1 )- G( x 2 ) - ( x 1 ) + ( x 2 ) = = 1 ò [ D G ( x 1 + x ( x 1 - x 2 )) - D ( x 1 + x( x 1 - x 2 ))] dx ( x 1 - x 2 ) 0 I follows from (45) ha H ( x 1, x 2 ) 1-- G x x 2 This equaion and inequaliy (46) imply he esimae Tx ( 1 )- Tx ( 2 ) 3-- x x 2 Therefore, he mapping T has a unique fixed poin in he ball B e = { x : x-y e} The lemma is proved Le he hypoheses of Theorems 41 and 42 hold We assume ha h < 1 in (41) Then for any elemen g Î W h ( f ) he following esimaes hold: gx ( ) M, Dg( x) M, x Î O, (47) where M > 0 is a consan In paricular, hese esimaes are valid for he mapping f Lemma 42 Le { y n : n Î Z} be a d-pseudorajecory of he mapping f lying in L Then for any k ³ 1 he sequence { z n º y nk : n Î Z} is a d M k -1 -pseudorajecory of he mapping f k Here M k has he form M k = 1 + M + ¼ + M k, k ³ 1, M 0 = 1, (48) and M is a consan from (47)

24 Hyperboliciy of Invarian Anosov s Ses for Lemma Differeniable on e -rajecories Mappings 385 Proof Le us use inducion o prove ha y nk+ i -f i ( y nk ) d M i -1, 1 i k (49) Since { y n } is a d-pseudorajecory, hen i is eviden ha for i = 1 inequaliy (49) is valid Assume ha equaion (49) is valid for some i ³ 1 and prove esimae (49) for i + 1 : y nk + i f i + 1 ( y nk ) y - nk + i + 1 f( y nk+ i ) + f ( y nk + ) - f( f i ( y i nk )) Wih he help of (47) we obain ha y nk + i f i + 1 ( y nk ) d+ M y nk+ i - f i ( y nk ) d+ M dm i -1 = d M i Thus, Lemma 42 is proved Lemma 43 Le { y n } be a d-pseudoorbi of he mapping f lying in L Le { x n } be a rajecory of he mapping g Î W h ( f ) such ha y nk -x nk e, n Î Z, (410) for some k ³ 1 If max( e, d+ h) M k D, (411) hen y n -x n max ( e, d+ h) M k, (412) where M k has he form (48) Proof We firs noe ha - = y nk 1 - y nk+ 1 x nk+ 1 ( ) + gx nk y nk+ 1 - f( y nk ) + f( y nk ) - gy ( nk ) + gy ( nk ) gx nk Therefore, i is eviden ha y nk+ 1 -x nk+ 1 d + h + M y nk - x nk max ( e, d+ h) ( 1 + M) (413) Here we use he esimae - ( ) 1 gy ( )- gx ( ) Dg( y + x ( x-y) ) dx x-y M x-y ò 0 which follows from (47) and holds when he segmen connecing he poins x and y lies in O Condiion (411) guaranees he fulfillmen of his propery a each sage of reasoning If we repea he argumens from he proof of (413), hen i is easy o complee he proof of (412) using inducion as in Lemma 42 Lemma 43 is proved

25 386 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 Lemma 44 Le y ÎL and x-y e Assume ha max ( e, h) ( 1 + ¼ + M k ) < D (414) Then he esimaes f j ( y) - f j ( x) M j x- y, Df j ( x) M j, (415) f j ( y) -g j ( x) max( e, h) ( 1 + ¼ + M j ), (416) f j ( x) -g j ( x) h( 1 + ¼ + M j -1 ) are valid for j = 1, 2, ¼, k and for every mapping g Î W h ( f ) (417) Proof As above, le us use inducion If j = 1, i is eviden ha equaions (415) (417) hold The ransiion from j o j + 1 in (415) is eviden Le us consider esimae (416): f j + 1 ( y) - g j + 1 ( x) = æf( f j ( y) ) - f( g j ( x) ) ö + f( g (418) è ø j ( x) ) - gg ( j ( x) ) Condiion (414) and he inducion assumpion give us ha g j ( x) lies in he ball wih he cenre a he poin f j ( y) Î L lying in O Therefore, i follows from (418) ha f j + 1 ( y) - g j + 1 ( x) M f j ( y) -g j ( x) + h max ( e, h) ( 1 + ¼ + M j + 1 ) The ransiion from j o j + 1 in (417) can be made in a similar way Lemma 44 is proved Lemma 45 There exiss D D such ha he equaions ì f k ( x) g k ü sup í - ( x) : x Î O ý r k ( h) (419) î þ and ì ( Df k )( x) Dg k ü sup í - ( )( x) : x Î O ý r k ( h) (420) î þ are valid in he D -viciniy O of he se L for any funcion g ÎW h ( f ) Here r k ( h) 0 as h 0 The proof follows from he definiion of he class of funcions W h ( f ) and esimaes (47) and (417)

26 Hyperboliciy of Invarian Anosov s Ses for Lemma Differeniable on e -rajecories Mappings 387 Le us also inroduce he values ì w k ( š) Df k ( y) Df k ü = sup í - ( x) : y Î L, x-y šý î þ (421) and ì ü wš ( ) = sup í Py ( )- P( x), x, y Î L, x-y šý î þ (422) The requiremen of he uniform coninuiy of he derivaive Df( x) (see he hypoheses of Theorem 41) and he projecors Px ( ) (see he hyperboliciy definiion) enables us o sae ha w k ( š) 0, w( š) 0 as š 0 (423) Le { y n } be a d-pseudorajecory of he mapping f lying in L Then due o Lemma 42 he sequence { y n = y nk : n Î Z} is a d M k -1 -pseudorajecory of he mapping f k Le us consider he mappings ( x) and G( x) in he space l X º l ( Z, X) (for he definiion see Secion 2) given by he equaliies [ ( x) ] n = y n + x n - f k ( y n -1 + x n -1 ), (424) [ G( x) ] n = y n + x n - g k ( y n -1 + x n -1 ), (425) where x = { x n : n Î Z} is an elemen from l X Thus, he consrucion of e -rajecories of he mapping f k and g k corresponding o he sequence { y n } is reduced o solving of he equaions ( x) = 0 and G( x) = 0 in he ball { x: x Le us show ha for large enough Lemma 41 can be lx e} k applied o he mappings and G Le us sar wih he mapping Lemma 46 The funcion is a C 1 -smooh mapping in l X wih he properies ( 0) dm k -1, (426) D ( x) -D ( 0) w k ( e), x (427) lx e Proof Esimae (426) follows from he fac ha { y n } is a d M k -1 -pseudorajecory Then i is eviden ha [ D ( x)h] n = h n - Df k ( y n -1 + x n -1 )h n -1, (428) where x = { x n : n Î Z} and h = { h n : n Î Z} lie in l X Therefore, simple calculaions and equaion (421) give us (427)

27 388 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 In order o deduce relaions (42) and (43) from inequaliies (426) and (427) for y = 0, we use Theorem 22 Consider he operaor L = D ( 0) I is clear ha [ L h] n = h n - Df k ( y n -1 )h n -1, h = { h n } Le us show ha equaions (29) (211) are valid for A n = Df k ( y n -1 ) and Q n = = P( y n -1 ) and hen esimae he corresponding consans Propery (29) follows from he hyperboliciy definiion Equaions (33) and (415) imply ha A n Q n Kq k and A n M k Furher, he permuabiliy propery (32) gives us ha Q n + 1 A n 1 Q n ( - ) = [ Q n + 1 A n - A n Q n ] Q n ) = = Q n P( f k ( y n -1 )) A n Q n ) Hence (see (422)), Q n + 1 A n Q n ) w( dm k -1 ) A n 1- Q n w( dm k -1 )M k K Similarly, we find ha Q n + 1 )A n Q n w( dm k -1 )M k K The operaor J n = Q n + 1 P( f k ( y n -1 )) - Q n + 1 ) P( f k ( y n -1 ))) is inverible if (see Lemma 31) Q n + 1 -P( f k ( y n -1 )) w( dm k -1 ) < K Moreover, J 1 n ( 1-2Kw ( d M k - )) 1 < 2, provided 2 Kw( dm k -1 ) < 1 2 Due o he hyperboliciy of he se L, he operaor A n is an inverible mapping from Q n )X ino P( f k ( y n -1 )))X Therefore, since Q n + 1 )A n Q n ) = J n A n Q n ), he operaor ( 1 - Q n + 1 )A n ( 1 - Q n ) is inverible as a mapping from Q n )X ino Q n + 1 )X Moreover, by virue of (34) we have ha [ Q n + 1 )A n ] ( 1 - Q n + ) 1 = [ A n Q n )] J 1 n ( 1 - Q n + ) 1 2 K 2 q k Thus, under he condiions 4 Kw( dm k -1 ) < 1, 8KM k w( dm k -1 ) 1, 16 K 3 q k 1, (429) Theorem 22 implies ha he operaor L = D ( 0) is inverible and L 2K + 1 Le us fix some e > 0 If

28 Hyperboliciy of Invarian Anosov s Ses for Lemma Differeniable on e -rajecories Mappings 389 dm k ( 4K + 2) e, w, 2 k ( e) ( 4K + 2) (430) hen by Lemma 46 relaions (42) and (43) hold wih y = 0, e = e, and q = 12 If r k ( h) 1-- min ( e, 1) ( 4K + 2), 2 (431) hen equaions (44) and (45) also hold wih y = 0, e = e, and q = 1 2 Hence, under condiions (429) (431) here exiss a unique soluion o equaion G( x) = 0 possessing he propery x l e This means ha for any d -pseudoorbi { y n : n Î Z} (lying in L) of he mapping f here exiss a unique rajecory { z n : n ÎZ} of he mapping g Î W h ( f ) such ha z nk -y nk e, n Î Z, provided condiions (429) (431) hold Therefore, under he addiional condiion ( e + d + h)m k D and due o Lemma 43 we ge y n -z n ( e + d + h)m k, n Î Z These properies are sufficien for he compleion of he proof of Theorem 42 Le us fix k such ha 16 K 3 q k 1 We choose e 0 D D ( D is defined in Lemma 45) such ha w k ( e) ( 4K + 2) e for all e M k Le us fix an arbirary e Î( 0, e 0 ] and ake e = e[ 2 M k ] Now we choose d = d( e) and h = h( e) such ha he following condiions hold: 4 Kw( dm k -1 ) < 1, 8KM k w( dm k -1 ) 1, dm k e ( 2K + 1), r, 4 k ( h) 1-- min ( e, 1) ( 2K + 1) 4 2 ( d+ h)m k e I is clear ha under such a choice of d and h any d-pseudoorbi (from L) of he mapping f has a unique e -rajecory of he mapping g Thus, Theorem 42 is proved Exercise 41 Le he hypoheses of Theorem 41 hold Show ha here exis D > 0 and d> 0 such ha for any wo rajecories { x n : n ÎZ} and { y n : n ÎZ} of a dynamical sysem ( X, f) he condiions dis( x n, L) < D, dis( y n, L) < D, sup x n - y n d n imply ha x n º y n, n Î Z In oher words, any wo rajecories of he sysem ( X, f) ha are close o a hyperbolic invarian se canno remain arbirarily close o each oher all he ime

29 390 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 Exercise 42 Show ha Theorem 41 admis he following srenghening: if he hypoheses of Theorem 41 hold, hen here exiss e 0 > 0 such ha for every e Î( 0, e 0 ) here exiss d = d( e) wih he propery ha for any d -pseudoorbi { y n } such ha dis( y n, L) < d here exiss a unique e -rajecory Exercise 43 Prove he analogue of he asserion of Exercise 42 for Theorem 42 Exercise 44 Le L = { x n : n Î Z} be a periodic orbi of he mapping f, ie f k ( x n ) º x n+ k = x n for all n Î Z and for some k ³ 1 Assume ha he hypoheses of Theorem 42 hold Then for h > 0 small enough every mapping g Î W h ( f ) possesses a periodic rajecory of he period k 5 Birkhoff-Smale Theorem One of he mos ineresing corollaries of Anosov s lemma is he Birkhoff-Smale heorem ha provides condiions under which he chaoic dynamics is observed in a discree dynamical sysem ( X, f) We remind (see Secion 1) ha by definiion he possibiliy of chaoic dynamics means ha here exiss an invarian se Y in he space X such ha he resricion of some degree f k of he mapping f on Y is opologically equivalen o he Bernoulli shif S in he space S m of wo-sided infinie sequences of m symbols Theorem 51 Le f be a coninuously differeniable mapping of a Banach space X ino iself Le x 0 Î X be a hyperbolic fixed poin of f and le { y n : n Î Z} be a homoclinic rajecory of he mapping f ha does no coincide wih x 0, ie f( x0 ) = x 0 ; f( y n ) = y n + 1, y n ¹ x 0, n Î Z ; y n x 0, n ± Assume ha he rajecory { y n : n Î Z} is ransversal, ie he se L = { x 0 } È { y n : n Î Z} is hyperbolic wih respec o f and here exiss a viciniy O of he se L such ha f( x) and Df( x) are bounded and uniformly coninuous on he closure O By W h ( f ) we denoe a se of coninuously differeniable map- pings g of he space X ino iself such ha f( x) -gx ( ) h, Df( x) -Dg( x) h, x ÎO

30 Birkhoff-Smale Theorem 391 Then here exiss h > 0 such ha for any mapping g Î W h ( f ) and for any m ³ 2 here exis a naural number l and a coninuous mapping j of he space S m ino a compac subse Y º js ( m ) in X such ha a) Y = js ( m ) is sricly invarian wih respec o g l, ie g l ( Y) = Y ; b) if a = ( ¼ a -1, a 0, a 1, ¼) and a = ( ¼a -1, a 0, a 1, ¼) are elemens of S m such ha a i ¹ a i for some i ³ 0, hen j( a) ¹ j( a ) ; c) he resricion of g l on Y is opologically conjugae o he Bernoulli shif S in S m, ie g l ( j( a) ) = j( Sa), a Î S m Moreover, if in addiion we assume ha for he mapping g here exiss e 0 > 0 such ha for any wo rajecories { x n : n Î Z} and { x n : n Î Z} (of he mapping g ) lying in he e 0 -viciniy of he se L he condiion x n0 = x n0 for some n 0 Î Z implies ha x n = x n for all n Î Z, hen he map- ping j is a homeomorphism The proof of his heorem is based on Anosov s lemma and mosly follows he sandard scheme (see, eg, [4]) used in he finie-dimensional case The only difficuly arising in he infinie-dimensional case is he proof of he coninuiy of he mapping j I can be overcome wih he help of he lemma presened below which is borrowed from he hesis by Jürgen Kalkbrenner (Augsburg, 1994) in fac I should also be noed ha he condiion under which j is a homeomorphism holds if he mapping g does no glue he poins in some viciniy of he se L, ie he equaliy gx ( ) = gx ( ) implies x = x Lemma 51 Le he hypoheses of Theorem 51 hold Le us inroduce he noaion J n º J n ( k 0, m) = { k Î Z : k-k 0 mn}, where k 0 ÎZ and m, n Î N Le z = { z n : n Î J n } be a segmen (lying in L) of a d- pseudoorbi of he mapping f : z n Î L, z n f( z n ) d, n, n+ 1 Î J n (51) Assume ha x = { x n : n Î J n } and x = { x n : n Î J n } are segmens of orbis of he mapping g Î W h ( f ): gx ( n ) = x n + 1, gx ( n ) = x n + 1, n, n+ 1 Î J n, (52) such ha z n -x n e, z n -x n e (53) Then here exis d, h, e > 0, and m Î N such ha condiions (51) (53) imply he inequaliy x k0-2 1 n x k0 - e (54)

31 392 Homoclinic Chaos in Infinie-Dimensional Sysems C h a p e r 6 Proof I follows from (52) ha where R k0 x k0 + m - x k0 + m = ( Df m )( z k0 )( x k0 - x k0 ) + R k0, (55) = g m ( x k0 ) - g m ( x k0 ) - ( Df m )( z k0 )( x k0 - x k0 ) Since he se L is hyperbolic wih respec o f, here exiss a family of projecors { Px ( ): x Î L} for which equaions (32) (34) are valid Therefore, P( f m ( z k0 )))( x k0 + m - x k0 + m) = = ( Df m )( z k0 ) Pz ( k0 ))( x k0 - x k0 ) + P( f m ( z k0 )))R k0 I means ha P( z k0 ))( x k0 - x k0 ) = = [( Df m )( z k0 )] -1 [ 1- P( f m ( z k0 ))][ x k0 + m - x k0 + m - R k0 ] Consequenly, equaion (34) implies ha Pz ( k0 ))( x k0 - x k0 ) Kq m ( x k0 + m - x k0 + m + R k0 ) Le us esimae he value R k0 I can be rewrien in he form 1 R k0 = [ ( Dg m )( xx k0 + x)x k0 ) - Df m ( z k0 ) ] dx ( x ò k0 - x k0 ) 0 I follows from (53) ha x x k0 + x)x k0 - z k0 e Hence, using (420) and (421), for e > 0 small enough we obain ha R k0 ( r m ( h) + w m ( e) ) x k0 - x k0, (56) where r m ( h) 0 as h 0 and w m ( e) 0 as e 0 Therefore, Pz ( k0 ))( x k0 - x k0 ) Kq m ( x k0 + m - x k0 + m + x k0 - x k0 ), (57) provided r m ( h) + w m ( e) 1 Furher, we subsiue he value k 0 - m for k 0 in (55) o obain ha x k0 - x k0 = Df m ( z k0 - m) ( x k0 - m - x k0 - m) + R k0 - m Therefore, using (32) we find ha P( f m ( z k0 -m ) )( x k0 - x k0 ) = = Df m( z k0 -m )Pz ( k0 - m )( x k0 -m - x k0 -m) + P ( f m ( z k0 -m) )R k0 - m

32 Birkhoff-Smale Theorem 393 Hence, equaions (33) and (56) wih k 0 -m insead of k 0 give us ha P( f m( z k0 -m ) )( x k0 - x k0 ) Since K{ q m + ( r m ( h) + w m ( e) )} x k0 -m - x k0 -m m -1 z k0 f m ì - ( z k0 -m) íf j ( z k0 - j) - f j ( f ( z k0 - j -1 ) ü = å ) ý, j = 0î þ i follows from (415) and (51) ha m -1 z k0 - f m ( z k0 - m) då M j d m( 1+ M m ) j = 0 for d small enough Therefore, Pz ( k0 ) -P( f m ( z k0 -m)) w( d m( 1 + M m )) º w( d, m), (58) where w( x) 0 as x 0 (cf (422)) Consequenly, esimae (58) implies ha P( z k0 )( x k0 - x k0 ) K{ q m + r m ( h) + w m ( e) + K -1 wd (, m) } x k0 -m - x k0 -m (59) I is eviden ha esimaes (57) and (59) enable us o choose he parameers m, h, e, and d such ha x k0 - x k0 1-- max { x 2 k - x k : k Î J 1 ( k 0, m) } Using his inequaliy wih k insead of k 0 we obain ha x k - x k 1-- max { x 2 n - x n : n Î J 2 ( k 0, m) } for all k Î J 1 ( k 0, m) Therefore, x k0 - x k0 1-- max { x 4 n - x n : n Î J 2 ( k 0, m) } If we coninue o argue like ha, hen we find ha x k0 - x k0 2 -n max { x n - x n : n Î J n ( k 0, m) } Since x n - x n x n - z n + x n - z n, his and esimae (53) imply (54) Lemma 51 is proved Proof of Theorem 51 Le p 1, p 2, ¼, p m -1 be disinc inegers Le us choose and fix he parameers e, h, d > 0 and he ineger m > 0 such ha (i) Theorem 42 and Lemma 51 can be applied o he hyperbolic se L = { x 0 } È { y n : n Î Z} and (ii)