«ACTA»00 Auho: Tile: I D Chueshov Inoducion o he Theoy of Infinie-Dimensionl Dissipive Sysems ISBN: 966 70 64 5 I D Chueshov Inoducion Theoy of Infinie-Dimensionl Dissipive Sysems o hetheoy Univesiylecuesinconempoymhemics You cn ORDER his book while visiing he websie of «ACTA» Scienific Publishing House hp://wwwccomu/en/ This book povides n exhu - sive inoducion o he scope of min ides nd mehods of he heoy of infinie-dimensionl dis - sipive dynmicl sysems which hs been pidly developing in e - cen yes In he exmples sys ems geneed by nonline pil diffeenil equions ising in he diffeen poblems of moden mechnics of coninu e consideed The min gol of he book is o help he ede o mse he bsic segies used in he sudy of infinie-dimensionl dissipive sysems nd o qulify him/he fo n independen scien - ific esech in he given bnch Expes in nonline dynmics will find mny fundmenl fcs in he convenien nd pcicl fom in his book The coe of he book is com - posed of he couses given by he uho he Depmen of Me chnics nd Mhemics Khkov Univesiy duing numbe of yes This book con - ins lge numbe of execises which mke he min ex moe complee I is sufficien o know he fundmenls of funcionl nlysis nd odiny diffeenil equions o ed he book Tnsled by Consnin I Chueshov fom he Russin ediion («ACTA», 999) Tnslion edied by Myn B Khoolsk
I D Chueshov Inoducion o he Theoy of Infinie-Dimensionl Dissipive Sysems A CTA 00
UDC 57 000 Mhemics Subjec Clssificion: pimy 37L05; secondy 37L30, 37L5 This book povides n exhusive inoducion o he scope of min ides nd mehods of he heoy of infinie-dimensionl dissipive dynmicl sysems which hs been pidly developing in ecen yes In he exmples sysems geneed by nonline pil diffeenil equions ising in he diffeen poblems of moden mechnics of coninu e consideed The min gol of he book is o help he ede o mse he bsic segies used in he sudy of infinie-dimensionl dissipive sysems nd o qulify him/he fo n independen scienific esech in he given bnch Expes in nonline dynmics will find mny fundmenl fcs in he convenien nd pcicl fom in his book The coe of he book is composed of he couses given by he uho he Depmen of Mechnics nd Mhemics Khkov Univesiy duing numbe of yes This book conins lge numbe of execises which mke he min ex moe complee I is sufficien o know he fundmenls of funcionl nlysis nd odiny diffeenil equions o ed he book Tnsled by Consnin I Chueshov fom he Russin ediion («ACTA», 999) Tnslion edied by Myn B Khoolsk wwwccomu ACTA Scienific Publishing House Khkiv, Ukine E-mil: we@ccomu I D Chueshov, 999, 00 Seies, «ACTA», 999 Typogphy, lyou, «ACTA», 00 ISBN 966-70-0-3 (seies) ISBN 966-70-64-5 Свідоцтво ДК 79
Conens Pefce 7 Chpe Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sys sems Noion of Dynmicl Sysem Tjecoies nd Invin Ses 7 3 Definiion of Aco 0 4 Dissipiviy nd Asympoic Compcness 4 5 Theoems on Exisence of Globl Aco 8 6 On he Sucue of Globl Aco 34 7 Sbiliy Popeies of Aco nd Reducion Pinciple 45 8 Finie Dimensionliy of Invin Ses 5 9 Exisence nd Popeies of Acos of Clss of Infinie-Dimensionl Dissipive Sysems 6 Refeences 73 Chpe Long-Time Behviou of Soluions o Clss of Semiline Pbolic Equions Posiive Opeos wih Discee Specum 77 Semiline Pbolic Equions in Hilbe Spce 85 3 Exmples 93 4 Exisence Condiions nd Popeies of Globl Aco 0 5 Sysems wih Lypunov Funcion 08 6 Explicily Solvble Model of Nonline Diffusion 8 7 Simplified Model of Appence of Tubulence in Fluid 30 8 On Reded Semiline Pbolic Equions 38 Refeences 45
4 Conens Chpe 3 Ineil Mnifolds Bsic Equion nd Concep of Ineil Mnifold 49 Inegl Equion fo Deeminion of Ineil Mnifold 55 3 Exisence nd Popeies of Ineil Mnifolds 6 4 Coninuous Dependence of Ineil Mnifold on Poblem Pmees 7 5 Exmples nd Discussion 76 6 Appoxime Ineil Mnifolds fo Semiline Pbolic Equions 8 7 Ineil Mnifold fo Second Ode in Time Equions 89 8 Appoxime Ineil Mnifolds fo Second Ode in Time Equions 00 9 Ide of Nonline Glekin Mehod 09 Refeences 4 Chpe 4 The Poblem on Nonline Oscillions of Ple in Supesonic Gs Flow Spces 8 Auxiliy Line Poblem 3 Theoem on he Exisence nd Uniqueness of Soluions 3 4 Smoohness of Soluions 40 5 Dissipiviy nd Asympoic Compcness 46 6 Globl Aco nd Ineil Ses 54 7 Condiions of Reguliy of Aco 6 8 On Singul Limi in he Poblem of Oscillions of Ple 68 9 On Ineil nd Appoxime Ineil Mnifolds 76 Refeences 8
Conens 5 Chpe 5 Theoy of Func cionls h Uniquely Deemine Long-Time Dynmics Concep of Se of Deemining Funcionls 85 Compleeness Defec 96 3 Esimes of Compleeness Defec in Sobolev Spces 306 4 Deemining Funcionls fo Absc Semiline Pbolic Equions 37 5 Deemining Funcionls fo Recion-Diffusion Sysems 38 6 Deemining Funcionls in he Poblem of Neve Impulse Tnsmission 339 7 Deemining Funcionls fo Second Ode in Time Equions 350 8 On Boundy Deemining Funcionls 358 Refeences 36 Chpe 6 Homoclinic Chos in Infinie-Dimensionl Sys sems Benoulli Shif s Model of Chos 365 Exponenil Dichoomy nd Diffeence Equions 369 3 Hypeboliciy of Invin Ses fo Diffeenible Mppings 377 4 Anosov s Lemm on e -jecoies 38 5 Bikhoff-Smle Theoem 390 6 Possibiliy of Chos in he Poblem of Nonline Oscillions of Ple 396 7 On he Exisence of Tnsvesl Homoclinic Tjecoies 40 Refeences 43 Index 45
Палкой щупая дорогу, Бродит наугад слепой, Осторожно ставит ногу И бормочет сам с собой И на бельмах у слепого Полный мир отображен: Дом, лужок, забор, корова, Клочья неба голубого Все, чего не видит он Вл Ходасевич «Слепой» A blind mn mps ndom ouching he od wih sick He plces his foo cefully nd mumbles o himself The whole wold is displyed in his ded eyes Thee e house, lwn, fence, cow nd scps of he blue sky eveyhing he cnno see Vl Khodsevich «A Blind Mn»
Pefce The ecen yes hve been mked ou by n evegowing inees in he esech of quliive behviou of soluions o nonline evoluiony pil diffeenil equions Such equions mosly ise s mhemicl models of pocesses h ke plce in el (physicl, chemicl, biologicl, ec) sysems whose ses cn be chceized by n infinie numbe of pmees in genel Dissipive sysems fom n impon clss of sysems obseved in eliy Thei min feue is he pesence of mechnisms of enegy ellocion nd dissipion Inecion of hese wo mechnisms cn led o ppence of compliced limi egimes nd sucues in he sysem Inense inees o he infinie-dimensionl dissipive sysems ws significnly simuled by emps o find deque mhemicl models fo he explnion of ubulence in liquids bsed on he noion of snge (iegul) co By now significn pogess in he sudy of dynmics of infinie-dimensionl dissipive sysems hve been mde Moeove, he les mhemicl sudies offe moe o less common line (segy), which when followed cn help o nswe numbe of pincipl quesions bou he popeies of limi egimes ising in he sysem unde consideion Alhough he mehods, ides nd conceps fom finie-dimensionl dynmicl sysems consiue he min souce of his segy, finie-dimensionl ppoches equie seious evluion nd dpion The book is devoed o sysemic inoducion o he scope of min ides, mehods nd poblems of he mhemicl heoy of infinie-dimensionl dissipive dynmicl sysems Min enion is pid o he sysems h e geneed by nonline pil diffeenil equions ising in he moden mechnics of coninu The min gol of he book is o help he ede o mse he bsic segies of he heoy nd o qulify him/he fo n independen scienific esech in he given bnch We lso hope h expes in nonline dynmics will find he fom mny fundmenl fcs e pesened in convenien nd pcicl The coe of he book is composed of he couses given by he uho he Depmen of Mechnics nd Mhemics Khkov Univesiy duing sevel yes The book consiss of 6 chpes Ech chpe coesponds o em couse (34-36 hous) ppoximely Is body cn be infeed fom he ble of conens Evey chpe includes sepe lis of efeences The efeences do no clim o be full The liss consis of he publicions efeed o in his book nd offe ddiionl woks ecommen-
8 Pefce ded fo fuhe eding Thee e lo of execises in he book They ply double ole On he one hnd, poofs of some semens e pesened s (o conin) cycles of execises On he ohe hnd, some execises conin n ddiionl infomion on he objec unde consideion We ecommend h he execises should be ed les Fomule nd semens hve double indexing in ech chpe (he fis digi is secion numbe) When fomule nd semens fom nohe chpe e efeed o, he numbe of he coesponding chpe is plced fis I is sufficien o know he bsic conceps nd fcs fom funcionl nlysis nd odiny diffeenil equions o ed he book I is quie undesndble fo unde-gdue sudens in Mhemics nd Physics ID Chueshov
Chpe Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems Conens Noion of Dynmicl Sysem Tjecoies nd Invin Ses 7 3 Definiion of Aco 0 4 Dissipiviy nd Asympoic Compcness 4 5 Theoems on Exisence of Globl Aco 8 6 On he Sucue of Globl Aco 34 7 Sbiliy Popeies of Aco nd Reducion Pinciple 45 8 Finie Dimensionliy of Invin Ses 5 9 Exisence nd Popeies of Acos of Clss of Infinie-Dimensionl Dissipive Sysems 6 Refeences 73
The mhemicl heoy of dynmicl sysems is bsed on he quliive heoy of odiny diffeenil equions he foundions of which wee lid by Heni Poincé (854 9) An essenil ole in is developmen ws lso plyed by he woks of A M Lypunov (857 98) nd A A Andonov (90 95) A pesen he heoy of dynmicl sysems is n inensively developing bnch of mhemics which is closely conneced o he heoy of diffeenil equions In his chpe we pesen some ides nd ppoches of he heoy of dynmicl sysems which e of genel-pupose use nd pplicble o he sysems geneed by nonline pil diffeenil equions Noion of Dynmicl Sysem In his book dynmicl sysem is ken o men he pi of objecs ( X, S ) consising of complee meic spce X nd fmily S of coninuous mppings of he spce X ino iself wih he popeies S + S S,, Î T +, S 0 = I, () whee T + coincides wih eihe se R + of nonnegive el numbes o se Z + = { 0,,, ¼} If T + = R +, we lso ssume h y() = S y is coninuous funcion wih espec o fo ny y Î X Theewih X is clled phse spce, o se spce, he fmily S is clled n evoluiony opeo (o semigoup), pmee Î T + plys he ole of ime If T + = Z +, hen dynmicl sysem is clled discee (o sysem wih discee ime) If T + = R +, hen ( X, S ) is fequenly clled o be dynmicl sysem wih coninuous ime If noion of dimension cn be defined fo he phse spce X (e g, if X is linel), he vlue dimx is clled dimension of dynmicl sysem Oiginlly dynmicl sysem ws undesood s n isoled mechnicl sysem he moion of which is descibed by he Newonin diffeenil equions nd which is chceized by finie se of genelized coodines nd velociies Now people ssocie ny ime-dependen pocess wih he noion of dynmicl sysem These pocesses cn be of quie diffeen oigins Dynmicl sysems nully ise in physics, chemisy, biology, economics nd sociology The noion of dynmicl sysem is he key nd uniing elemen in synegeics Is usge enbles us o cove he wide specum of poblems ising in picul sciences nd o wok ou univesl ppoches o he descipion of quliive picue of el phenomen in he univese
Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e Le us look he following exmples of dynmicl sysems Exmple Le f( x) be coninuously diffeenible funcion on he el xis posessing he popey xf( x) ³ -C ( + x ), whee C is consn Conside he Cuchy poblem fo n odiny diffeenil equion x () = -f( x() ), > 0, x ( 0) = x 0 () Fo ny x Î R poblem () is uniquely solvble nd deemines dynmicl sysem in R The evoluiony opeo S is given by he fomul S x 0 = x(), whee x() is soluion o poblem () Semigoup popey () holds by viue of he heoem of uniqueness of soluions o poblem () Equions of he ype () e ofen used in he modeling of some ecologicl pocesses Fo exmple, if we ke f( x) = x( x - ), > 0, hen we ge logisic equion h descibes gowh of populion wih compeiion (he vlue x() is he populion level; we should ke R + fo he phse spce) Exmple Le f( x) nd g( x) be coninuously diffeenible funcions such h x F( x) = f ( x) d x ³ -c, g( x) ³ -c 0 wih some consn c Le us conside he Cuchy poblem ì x + g( x)x + f( x) = 0, > 0, í (3) x ( 0) = x 0, x ( 0) = x Fo ny y 0 = ( x 0, x ) Î R, poblem (3) is uniquely solvble I genees wo-dimensionl dynmicl sysem ( R, S ), povided he evoluiony opeo is defined by he fomul S ( x 0 ; x ) = ( x(); x () ), whee x() is he soluion o poblem (3) I should be noed h equions of he ype (3) e known s Liénd equions in lieue The vn de Pol equion: nd he Duffing equion: gx ( ) = e( x - ), e > 0, f( x) = x gx ( ) = e, e > 0, f( x) = x 3 - x - b which ofen occu in pplicions, belong o his clss of equions
Noion of Dynmicl Sysem 3 Exmple 3 Le us now conside n uonomous sysem of odiny diffeenil equions x k () = f k ( x, x, ¼, x N), k =,, ¼, N (4) Le he Cuchy poblem fo he sysem of equions (4) be uniquely solvble ove n biy ime inevl fo ny iniil condiion Assume h soluion coninuously depends on he iniil d Then equions (4) genee n N - dimensionl dynmicl sysem ( R N, S ) wih he evoluiony opeo S cing in ccodnce wih he fomul S y 0 = ( x (), ¼, x N () ), y 0 = ( x 0, x 0, ¼, x N 0 ), whee { x i () } is he soluion o he sysem of equions (4) such h x i ( 0) = x i 0, i =,, ¼, N Genelly, le X be line spce nd F be coninuous mpping of X ino iself Then he Cuchy poblem x () = F( x() ), > 0, x ( 0) = x 0 Î X (5) genees dynmicl sysem ( X, S ) in nul wy povided his poblem is well-posed, ie heoems on exisence, uniqueness nd coninuous dependence of soluions on he iniil condiions e vlid fo (5) Exmple 4 Le us conside n odiny eded diffeenil equion x () + x() = f( x( - ) ), > 0, (6) whee f is coninuous funcion on R, > 0 Obviously n iniil condiion fo (6) should be given in he fom x() Î [-, 0] = f() (7) Assume h f () lies in he spce C [-, 0] of coninuous funcions on he segmen [-, 0] In his cse he soluion o poblem (6) nd (7) cn be consuced by sep-by-sep inegion Fo exmple, if 0, he soluion x() is given by x() = e f ( 0) + e ) f ( f( -) ) d 0 nd if Î [, ], hen he soluion is expessed by he simil fomul in ems of he vlues of he funcion x() fo Î [ 0, ] nd so on I is cle h he soluion is uniquely deemined by he iniil funcion f () If we now define n opeo S in he spce X = C[ -, 0] by he fomul ( S f) ( ) = x( + ), Î [-, 0], whee x() is he soluion o poblem (6) nd (7), hen we obin n infinie-dimensionl dynmicl sysem ( C [-, 0], S ),
4 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e Now we give sevel exmples of discee dynmicl sysems Fis of ll i should be noed h ny sysem ( X, S ) wih coninuous ime genees discee sysem if we ke Î Z + insed of Î R + Fuhemoe, he evoluiony opeo S of discee dynmicl sysem is degee of he mpping S, i e S = S, Î Z+ Thus, dynmicl sysem wih discee ime is deemined by coninuous mpping of he phse spce X ino iself Moeove, discee dynmicl sysem is vey ofen defined s pi ( X, S), consising of he meic spce X nd he coninuous mpping S Exmple 5 Le us conside one-sep diffeence scheme fo poblem (5): x n + - x n = F( x n ), n = 0,,, ¼, > 0 Thee ises discee dynmicl sysem ( X, S n ), whee S is he coninuous mpping of X ino iself defined by he fomul Sx = x + F( x) Exmple 6 Le us conside nonuonomous odiny diffeenil equion x () = f( x, ), > 0, x Î R, (9) whee f( x, ) is coninuously diffeenible funcion of is vibles nd is peiodic wih espec o, i e f( x, ) = f( x, + T) fo some T > 0 I is ssumed h he Cuchy poblem fo (9) is uniquely solvble on ny ime inevl We define monodomy opeo ( peiod mpping) by he fomul Sx 0 = x( T), whee x() is he soluion o (9) sisfying he iniil condiion x ( 0) = x 0 I is obvious h his opeo possesses he popey S k x() = x( + kt) (0) fo ny soluion x() o equion (9) nd ny k Î Z + The ising dynmicl sysem ( R, S k ) plys n impon ole in he sudy of he long-ime popeies of soluions o poblem (9) Exmple 7 (Benoulli shif) Le X = S be se of sequences x = { x i, i Î Z} consising of zeoes nd ones Le us mke his se ino meic spce by defining he disnce by he fomul d( x, y) = inf { -n : x i = y i, i < n} Le S be he shif opeo on X, i e he mpping nsfoming he sequence x = { x i } ino he elemen y = { y i }, whee y i = x i + As esul, dynmicl sysem ( X, S n ) comes ino being I is used fo descibing compliced (qusindom) behviou in some quie elisic sysems
Noion of Dynmicl Sysem 5 In he exmple below we descibe one of he ppoches h enbles us o connec dynmicl sysems o nonuonomous (nd nonpeiodic) odiny diffeenil equions Exmple 8 Le h( x, ) be coninuous bounded funcion on R Le us define he hull L h of he funcion h( x, ) s he closue of se ì ü íh ( x, ) º h( x, + ), Î Rý þ wih espec o he nom ì ü h C = sup íh( x, ) : x Î R, Î Rý þ Le gx ( ) be coninuous funcion I is ssumed h he Cuchy poblem x () = gx ( ) + h ( x, ), x ( 0) = x 0 () is uniquely solvble ove he inevl [ 0, + ) fo ny h Î L h Le us define he evoluiony opeo S on he spce X = R L h by he fomul S ( x 0, h ) = ( x ( ), h ), whee x() is he soluion o poblem () nd h = h ( x, + ) As esul, dynmicl sysem ( R L h, S ) comes ino being A simil consucion is ofen used when L h is compc se in he spce C of coninuous bounded funcions (fo exmple, if h( x, ) is qusipeiodic o lmos peiodic funcion) As he following exmple shows, his ppoch lso enbles us o use nully he noion of he dynmicl sysem fo he descipion of he evoluion of objecs subjeced o ndom influences Exmple 9 Assume h f 0 nd f e coninuous mppings fom meic spce Y ino iself Le Y be se spce of sysem h evolves s follows: if y is he se of he sysem ime k, hen is se ime k + is eihe f 0 ( y) o f ( y) wih pobbiliy, whee he choice of f 0 o f does no depend on ime nd he pevious ses The se of he sysem cn be defined fe numbe of seps in ime if we flip coin nd wie down he sequence of evens fom he igh o he lef using 0 nd Fo exmple, le us ssume h fe 8 flips we ge he following se of oucomes: ¼ 0 000 Hee coesponds o he hed flling, whees 0 coesponds o he il flling Theewih he se of he sysem ime = 8 will be wien in he fom:
6 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e W = ( f f 0 f f f 0 f 0 f f 0) ( y) This consucion cn be fomlized s follows Le S be se of wo-sided sequences consising of zeoes nd ones (s in Exmple 7), ie collecion of elemens of he ype w = ( ¼w -n ¼w - w 0 w ¼w n ¼ ), whee w i is equl o eihe o 0 Le us conside he spce X = S Y consising of pis x = ( w, y), whee w Î S, y Î Y Le us define he mpping F: X X by he fomul: F( x) º F( w, y) = ( S w, f w0 ( y) ), whee S is he lef-shif opeo in S (see Exmple 7) I is esy o see h he n - h degee of he mpping F ccs ccoding o he fomul F n ( w, y) = ( S n w, ( f wn - ¼ f w f w )( y) ) 0 nd i genees discee dynmicl sysem ( S Y, F n ) This sysem is ofen clled univesl ndom (discee) dynmicl sysem Exmples of dynmicl sysems geneed by pil diffeenil equions will be given in he chpes o follow Execise Assume h opeos S hve coninuous invese fo ny Show h he fmily of opeos { S : Î R} defined by he equliy S = S fo ³ 0 nd S - = S fo < 0 fom goup, ie () holds fo ll, Î R Execise Pove he unique solvbiliy of poblems () nd (3) involved in Exmples nd Execise 3 Gound fomul (0) in Exmple 6 in Exmple 8 possesses semi- Execise 4 Show h he mpping goup popey () S Execise 5 Show h he vlue d( x, y) involved in Exmple 7 is meic Pove is equivlence o he meic d * ( x, y) = å i x i - y i i = -
Tjecoies nd Invin Ses 7 Tjecoies nd Invin Ses Le ( X, S ) be dynmicl sysem wih coninuous o discee ime Is jecoy (o obi) is defined s se of he ype g = { u(): Î T}, whee u() is coninuous funcion wih vlues in X such h S u() = u( + ) fo ll Î T + nd Î T Posiive (negive) semijecoy is defined s se g + = { u(): ³ 0}, ( g = { u(): 0}, especively), whee coninuous on T + ( T, especively) funcion u() possesses he popey S u() = u( + ) fo ny > 0, ³ 0 ( > 0, 0, + 0, especively) I is cle h ny posiive semijecoy g + hs he fom g + = { S v : ³ 0}, ie i is uniquely deemined by is iniil se v To emphsize his cicumsnce, we ofen wie g + = g + ( v) In genel, i is impossible o coninue his semijecoy g + ( v) o full jecoy wihou imposing ny ddiionl condiions on he dynmicl sysem Execise Assume h n evoluiony opeo S is inveible fo some > 0 Then i is inveible fo ll > 0 nd fo ny v Î X hee exiss negive semijecoy g = g ( v) ending he poin v A jecoy g = { u(): Î T} is clled peiodic jecoy (o cycle) if hee exiss T Î T +, T > 0 such h u( + T) = u() Theewih he miniml numbe T > 0 possessing he popey menioned bove is clled peiod of jecoy Hee T is eihe R o Z depending on whehe he sysem is coninuous o discee one An elemen u 0 Î X is clled fixed poin of dynmicl sysem ( X, S ) if S u 0 = u 0 fo ll ³ 0 (synonyms: equilibium poin, siony poin) Execise Find ll he fixed poins of he dynmicl sysem ( R, S ) geneed by equion () wih f( x) = x( x - ) Does hee exis peiodic jecoy of his sysem? Execise 3 Find ll he fixed poins nd peiodic jecoies of dynmicl sysem in R geneed by he equions ìx = - y - x[ ( x + y ) - 4 ( x + y ) + ], ï í ïy = x - y[ ( x + y ) - 4 ( x + y ) + ] Conside he cses ¹ 0 nd = 0 Hin: use pol coodines Execise 4 Pove he exisence of siony poins nd peiodic jecoies of ny peiod fo he discee dynmicl sysem descibed
8 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e in Exmple 7 Show h he se of ll peiodic jecoies is dense in he phse spce of his sysem Mke sue h hee exiss jecoy h psses wheve smll disnce fom ny poin of he phse spce The noion of invin se plys n impon ole in he heoy of dynmicl sysems A subse Y of he phse spce X is sid o be: ) posiively invin, if S Y Í Y fo ll ³ 0 ; b) negively invin, if S Y Ê Y fo ll ³ 0 ; c) invin, if i is boh posiively nd negively invin, ie if S Y = Y fo ll ³ 0 The simples exmples of invin ses e jecoies nd semijecoies Execise 5 Show h g + is posiively invin, g is negively invin nd g is invin Execise 6 Le us define he ses nd g + ( A) = S ( A) º v= S ³È u : u Î A ³È 0 0 { } g ( A) = S- ³È ( A) º { v : S v Î A} 0 0 fo ny subse A of he phse spce X Pove h g + ( A) is posiively invin se, nd if he opeo S is inveible fo some > 0, hen g ( A) is negively invin se Ohe impon exmple of invin se is conneced wih he noions of w -limi nd -limi ses h ply n essenil ole in he sudy of he long-ime behviou of dynmicl sysems Le A Ì X Then he w-limi se fo A is defined by w ( A) = S, ³È ( A) s 0 s X whee S ( A) = { v= S u : u Î A} Heeinfe [ Y] X is he closue of se Y in he spce X The se s 0 s - whee S ( A) = { v : S v Î A}, is clled he -limi se fo A ³Ç ( A) = S- ( A) ³Ç ³È ³È X,
Tjecoies nd Invin Ses 9 Lemm n Fo n elemen y o belong o n w-limi se w( A), i is necessy nd sufficien h hee exis sequence of elemens { y n } Ì A nd sequence of numbes, he le ending o infiniy such h lim d Sn yn, y) =, n whee d( x, y) is he disnce beween he elemens x nd y in he spce X Poof Le he sequences menioned bove exis Then i is obvious h fo ny > 0 hee exiss n 0 ³ 0 such h This implies h S n y n Î S ³È ( A), n ³ n 0 y= lim S n y n Î S ( ) n X ³È A fo ll > 0 Hence, he elemen y belongs o he inesecion of hese ses, ie y Î w ( A) On he cony, if y Î w ( A), hen fo ll n = 0,,, ¼ y Î S ³È ( A) n X Hence, fo ny n hee exiss n elemen z n such h z n Î S ³È ( A), d( y, z n ) n n Theewih i is obvious h z n = S n y n, y n Î A, n ³ n This poves he lemm I should be noed h his lemm gives us descipion of n w -limi se bu does no gunee is nonempiness Execise 7 Show h w ( A) is posiively invin se If fo ny > 0 hee exiss coninuous invese o S, hen w ( A) is invin, ie S w ( A) = w( A) Execise 8 Le S be n inveible mpping fo evey > 0 Pove he counep of Lemm fo n -limi se: ì - ü y Î ( A) Û í$ { y n } Î A, $ n, n + ; lim d( S n yn, y) = 0ý n þ Esblish he invince of ( A)
0 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e Execise 9 Le g = { u(): - < < } be peiodic jecoy of dynmicl sysem Show h g = w( u) = ( u) fo ny u Î g Execise 0 Le us conside he dynmicl sysem ( R, S ) consuced in Exmple Le nd b be he oos of he funcion f( x): f( ) = f( b) = 0, < b Then he segmen I = { x: x b} is n invin se Le F( x) be pimiive of he funcion f( x) ( F ( x) = f( x) ) Then he se { x : Fx ( ) c} is posiively invin fo ny c Execise Assume h fo coninuous dynmicl sysem ( X, S ) hee exiss coninuous scl funcion V( y) on X such h he vlue V( S y) is diffeenible wih espec o fo ny y Î X nd d ( VS (, d y) ) + VS ( y) ( > 0, > 0, y Î X) Then he se ³ { y : Vy ( ) R} is posiively invin fo ny R ³ 3 Definiion of Aco co Aco is cenl objec in he sudy of he limi egimes of dynmicl sysems Sevel definiions of his noion e vilble Some of hem e given below Fom he poin of view of infinie-dimensionl sysems he mos convenien concep is h of he globl co A bounded closed se A Ì X is clled globl co fo dynmicl sysem ( X, S ), if ) A is n invin se, ie S A = A fo ny > 0 ; ) he se A unifomly cs ll jecoies sing in bounded ses, ie fo ny bounded se B fom X ì ü lim sup ídis ( S y, A ): y Î Bý = 0 þ We emind h he disnce beween n elemen z nd se A is defined by he equliy: dis ( z, A) = inf { d( z, y): y Î A}, whee d( z, y) is he disnce beween he elemens z nd y in X The noion of wek globl co is useful fo he sudy of dynmicl sysems geneed by pil diffeenil equions
Definiion of Aco Le X be complee line meic spce A bounded wekly closed se A is clled globl wek co if i is invin ( S A = A, > 0) nd fo ny wek viciniy O of he se A nd fo evey bounded se B Ì X hee exiss 0 = 0 ( O, B) such h S B Ì O fo ³ 0 We emind h n open se in wek opology of he spce X cn be descibed s finie inesecion nd subsequen biy union of ses of he fom U l, c = { x Î X: l( x) < c}, whee c is el numbe nd l is coninuous line funcionl on X I is cle h he conceps of globl nd globl wek cos coincide in he finie-dimensionl cse In genel, globl co A is lso globl wek co, povided he se A is wekly closed Execise 3 Le A be globl o globl wek co of dynmicl sysem ( X, S ) Then i is uniquely deemined nd conins ny bounded negively invin se The co A lso conins he w - limi se w ( B) of ny bounded B Ì X Execise 3 Assume h dynmicl sysem ( X, S ) wih coninuous ime possesses globl co A Le us conside discee sysem ( X, T n ), whee T = S 0 wih some 0 > 0 Pove h A is globl co fo he sysem ( X, T n ) Give n exmple which shows h he convese sseion does no hold in genel If he globl co A exiss, hen i conins globl miniml co A 3 which is defined s miniml closed posiively invin se possessing he popey lim dis ( S y, A 3 ) = 0 fo evey y Î X By definiion minimliy mens h A 3 hs no pope subse possessing he popeies menioned bove I should be noed h in cons wih he definiion of he globl co he unifom convegence of jecoies o A 3 is no expeced hee Execise 33 Show h S A 3 = A 3, povided A 3 is compc se Execise 34 Pove h w ( x) Î A 3 fo ny x Î X Theewih, if A 3 is compc, hen A 3 = È { w ( x ): x Î X} By definiion he co A 3A3 conins limi egimes of ech individul jecoy I will be shown below h ¹ A in genel Thus, se of el limi egimes (ses) oigining in dynmicl sysem cn ppe o be nowe hn he globl co Moeove, in some cses some of he ses h e unessenil fom he poin of view of he fequency of hei ppence cn lso be emoved fom A 3, fo exmple, such ses like bsoluely unsble siony poins The nex wo definiions ke ino ccoun he fc menioned bove Unfounely, hey equie
Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e ddiionl ssumpions on he popeies of he phse spce Theefoe, hese definiions e mosly used in he cse of finie-dimensionl dynmicl sysems Le Boel mesue m such h m( X) < be given on he phse spce X of dynmicl sysem ( X, S ) A bounded se A 4 in X is clled Milno co (wih espec o he mesue m ) fo ( X, S ) if A 4 is miniml closed invin se possessing he popey lim dis ( S y, A 4 ) = 0 fo lmos ll elemens y ÎX wih espec o he mesue m The Milno co is fequenly clled pobbilisic globl miniml co A ls le us inoduce he noion of sisiclly essenil globl miniml co suggesed by Ilyshenko Le U be n open se in X nd le X U ( x) be is chceisic funcion: X U ( x) =, x Î U; X U ( x) = 0, x Ï U Le us define he vege ime ( x, U) which is spen by he semijecoy g + ( x) emning fom x in he se U by he fomul T ( x, U) = lim T T X U ( S x) d 0 A se U is sid o be unessenil wih espec o he mesue m if M( U) º m{ x : ( x, U) > 0} = 0 The complemen A 5 o he mximl unessenil open se is clled n Ilyshenko co co (wih espec o he mesue m ) I should be noed h he cos A 4 nd A 5 e used in cses when he nul Boel mesue is given on he phse spce (fo exmple, if X is closed mesuble se in R N nd m is he Lebesgue mesue) The elions beween he noions inoduced bove cn be illused by he following exmple Exmple 3 Le us conside qusi-hmilonin sysem of equions in R : ì ïq = ï í ï ïp = H H - m H, p q - H q mh H -, p (3) whee H( p, q) = ( )p + q 4 - q nd m is posiive numbe I is esy o scein h he phse poi of he dynmicl sysem geneed by equions (3) hs he fom epesened on Fig
Definiion of Aco 3 A sepix ( eigh cuve ) sepes he domins of he phse plne wih he diffeen quliive behviou of he jecoies I is given by he equion Hpq (, ) = 0 The poins ( p, q) inside he sepix e chceized by he equion H( p, q) < 0 Theewih i ppes h Fig Phse poi of sysem (3) A = A = {( p, q): H( p, q) 0}, ì ü ì A 3 í( p, q): H( p, q) = 0ý ( p, q): H( p, q) ü = È í = H( p, q) = 0 ý þ p q þ A 4 = {( p, q): H( p, q) = 0} A 5 = { 0, 0}, ie he Ilyshenko - Finlly, he simple clculions show h co consiss of single poin Thus, ll inclusions being sic A = A É A 3 É A 4 É A 5,, Execise 35 Disply gphiclly he cos A j by equions (3) on he phse plne of he sysem geneed Execise 36 Conside he dynmicl sysem fom Exmple wih f( x) = x( x - ) Pove h A = { x : - x }, A 3 = { x = 0 ; x =± }, nd A 4 = A 5 = { x = ± } Execise 37 Pove h A 4 Ì A 3 nd A 5 Ì A 3 in genel Execise 38 Show h ll posiive semijecoies of dynmicl sysem which possesses globl miniml co e bounded ses In picul, he esul of he ls execise shows h he globl co cn exis only unde ddiionl condiions concening he behviou of jecoies of he sysem infiniy The min condiion o be me is he dissipiviy discussed in he nex secion
4 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e 4 Dissipiviy nd Asympoic Compcness Fom he physicl poin of view dissipive sysems e pimily conneced wih ievesible pocesses They epesen he wide nd impon clss of he dynmicl sysems h e inensively sudied by moden nul sciences These sysems (unlike he consevive sysems) e chceized by he exisence of he ccened diecion of ime s well s by he enegy ellocion nd dissipion In picul, his mens h limi egimes h e siony in cein sense cn ise in he sysem when + Mhemiclly hese feues of he quliive behviou of he jecoies e conneced wih he exisence of bounded bsobing se in he phse spce of he sysem A se B 0 Ì X is sid o be bsobing fo dynmicl sysem ( X, S ) if fo ny bounded se B in X hee exiss 0 = 0 ( B) such h S ( B) Ì B 0 fo evey ³ 0 A dynmicl sysem ( X, S ) is sid o be dissipive if i possesses bounded bsobing se In cses when he phse spce X of dissipive sysem ( X, S ) is Bnch spce bll of he fom { x Î X: x X R} cn be ken s n bsobing se Theewih he vlue R is sid o be dius of dissipiviy As ule, dissipiviy of dynmicl sysem cn be deived fom he exisence of Lypunov ype funcion on he phse spce Fo exmple, we hve he following sseion Theoem 4 Le he phse spce s of coninuous dynmicl sysem ( X, S ) be B- nch spce Assume h: () hee exiss coninuous funcion U( x) on X possessing he po- peies j ( x ) U( x) j ( x ), (4) whee j j ( ) e coninuous funcions on R + nd j ( ) + when ; (b) hee exis deivive d US ( d y) nd such h fo ³ 0 nd posiive numbes d US ( d y) - fo S y > (4) Then he dynmicl sysem ( X, S ) is dissipive Poof Le us choose R 0 ³ such h j ( ) > 0 fo ³ R 0 Le l = sup { j ( ): + R 0 } nd R > R 0 + be such h j ( ) > l fo > R Le us show h
Dissipiviy nd Asympoic Compcness 5 S y R fo ll ³ 0 nd y R 0 (43) Assume he cony, ie ssume h fo some y Î X such h y R 0 hee exiss ime > 0 possessing he popey S y > R Then he coninuiy of S y implies h hee exiss 0 < 0 < such h < S 0 y R 0 + Thus, equion (4) implies h US ( y) US ( 0 y), ³ 0, povided S y > I follows h US ( y) l fo ll ³ 0 Hence, S y R fo ll ³ 0 This condics he ssumpion Le us ssume now h B is n biy bounded se in X h does no lie inside he bll wih he dius R 0 Then equion (4) implies h US ( y) Uy ( ) - l B -, y Î B, (44) povided S y > Hee l B = sup { Ux ( ): x Î B} Le y Î B If fo ime * < ( l B - l) he semijecoy S y enes he bll wih he dius, hen by (43) we hve S y R fo ll ³ * If h does no ke plce, fom equion (44) i follows h j ( S y ) US ( y) l fo l B - l ³, ie S y R fo ³ ( l B - l) Thus, S B Ì { x: x R }, l B - l ³ This nd (43) imply h he bll wih he dius R is n bsobing se fo he dynmicl sysem ( X, S ) Thus, Theoem 4 is poved Execise 4 Show h hypohesis (4) of Theoem 4 cn be eplced by he equiemen d US ( d y) + gus ( y) C, whee g nd C e posiive consns Execise 4 Show h he dynmicl sysem geneed in R by he diffeenil equion x + f( x) = 0 (see Exmple ) is dissipive, povided he funcion f( x) possesses he popey: xf( x) ³ dx - C, whee d > 0 nd C e consns (Hin: U( x) = x ) Find n uppe esime fo he miniml dius of dissipiviy Execise 43 Conside discee dynmicl sysem ( R, f n ), whee f is coninuous funcion on R Show h he sysem ( R, f) is dissipive, povided hee exis > 0 nd 0 < < such h f( x) < x fo x >
6 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e Execise 44 Conside dynmicl sysem ( R, S ) geneed (see Exmple ) by he Duffing equion x + e x + x 3 - x = b, whee nd b e el numbes nd e > 0 Using he popeies of he funcion Ux (, x ) = x + x 4 4 - x + n æxx + e xö show h he dynmicl sysem ( R, S ) is dissipive fo n > 0 smll enough Find n uppe esime fo he miniml dius of dissipiviy Execise 45 Pove he dissipiviy of he dynmicl sysem geneed by (4) (see Exmple 3), povided N N x k f ( k x, x, ¼, x ) å N -d x k + C, k = k å d > 0 = Execise 46 Show h he dynmicl sysem of Exmple 4 is dissipive if f( z) is bounded funcion Execise 47 Conside cylinde Ц wih coodines ( x, j), x Î R, j Î [ 0, ) nd he mpping T of his cylinde which is defined by he fomul T( x, j) = ( x, j ), whee x = x + ksin pj, j = j + x ( mod) Hee nd k e posiive pmees Pove h he discee dynmicl sysem ( Ц, T n ) is dissipive, povided 0 < < We noe h if =, hen he mpping T is known s he Chiikov mpping I ppes in some poblems of physics of elemeny picles Execise 48 Using Theoem 4 pove h he dynmicl sysem ( R, S ) geneed by equions (3) (see Exmple 3) is dissipive (Hin: Ux ( ) = [ H( p, q) ] ) In he poof of he exisence of globl cos of infinie-dimensionl dissipive dynmicl sysems ge ole is plyed by he popey of sympoic compcness Fo he ske of simpliciy le us ssume h X is closed subse of Bnch spce The dynmicl sysem ( X, S ) is sid o be sympoiclly compc if fo ny > 0 is evoluiony opeo S cn be expessed by he fom ( ) ( ) S = S + S, (45) ( ) ( ) whee he mppings S nd S possess he popeies:
Dissipiviy nd Asympoic Compcness 7 ) fo ny bounded se B in X ( ) B () = sup S y X 0, + ; y ÎB b) fo ny bounded se B in X hee exiss 0 such h he se ( ) [ g 0 ( B) ] = ( ) S 0 ³È B (46) is compc in X, whee [ g] is he closue of he se g A dynmicl sysem is sid o be compc if i is sympoiclly compc nd ( ) one cn ke S º 0 in epesenion (45) I becomes cle h ny finie-dimensionl dissipive sysem is compc Execise 49 Show h condiion (46) is fulfilled if hee exiss compc ( ) se K in H such h fo ny bounded se B he inclusion S B Ì K, ³ 0 ( B) holds In picul, dissipive sysem is compc if i possesses compc bsobing se Lemm 4 The dynmicl sysem ( X, S ) is sympoiclly compc if hee exiss compc se K such h lim sup { dis ( S u, K) : u Î B} = 0 (47) fo ny se B bounded in X Poof The disnce o compc se is eched on some elemen Hence, fo ny > 0 nd u Î X hee exiss n elemen v º ( ) S u Î K such h dis ( S u, K) = S u - ( ) S u ( ) Theefoe, if we ke S u = S u - ( ) S u, i is esy o see h in his cse decomposiion (45) sisfies ll he equiemens of he definiion of sympoic compcness Remk 4 In mos pplicions Lemm 4 plys mjo ole in he poof of he popey of sympoic compcness Moeove, in cses when he phse spce X of he dynmicl sysem ( X, S ) does no possess he sucue of line spce i is convenien o define he noion of he sympoic compcness using equion (47) Nmely, he sysem ( X, S ) is sid o be sympoiclly compc if hee exiss compc K possessing popey (47) fo ny bounded se B in X Fo one moe ppoch o he definiion of his concep see Execise 5 below
8 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e Execise 40 Conside he infinie-dimensionl dynmicl sysem geneed by he eded equion x () + x() = f( x( -) ), whee > 0 nd f( z) is bounded (see Exmple 4) Show h his sysem is compc Execise 4 Conside he sysem of Loenz equions ising s heemode Glekin ppoximion in he poblem of convecion in hin lye of liquid: ì x = -s x + sy, ï í y = x - y - xz, ï z = - bz + xy Hee s,, nd b e posiive numbes Pove he dissipiviy of he dynmicl sysem geneed by hese equions in R 3 Hin: Conside he funcion V( x, y, z) on he jecoies of he sysem = ( x + y +( z - - s) ) 5 Theoems on Exisence of Globl Aco co Fo he ske of simpliciy i is ssumed in his secion h he phse spce X is Bnch spce, lhough he min esuls e vlid fo wide clss of spces (see, e g, Execise 58) The following sseion is he min esul Theoem 5 Assume h dynmicl sysem ( X, S ) is dissipive nd sympoi- clly compc Le B be bounded bsobing se of he sysem ( X, S ) Then he se A = w( B) is nonempy compc se nd is globl co of he dynmicl sysem ( X, S ) The co A is conneced se in X In picul, his heoem is pplicble o he dynmicl sysems fom Execises 4 4 I should lso be noed h Theoem 5 long wih Lemm 4 gives he following cieion: dissipive dynmicl sysem possesses compc globl co if nd only if i is sympoiclly compc The poof of he heoem is bsed on he following lemm
Theoems on Exisence of Globl Aco 9 Lemm 5 Le dynmicl sysem ( X, S ) be sympoiclly compc Then fo ny bounded se B of X he w-limi se w( B) is nonempy compc invin se Poof ( ) Le y n ÎB Then fo ny sequence { n } ending o infiniy he se { S n yn, n =,, ¼} is elively compc, ie hee exis sequence n k nd n elemen y Î X such h S nk ynk ends o y s k Hence, he sympoic ( ) compcness gives us h ( ) ( ) y - S nk y nk S nk ynk + y - S n ynk 0 s k Thus, y = lim S nk y nk Due o Lemm his indices h w( B) is nonempy k Le us pove he invince of w -limi se Le y Î w ( B) Then ccoding o Lemm hee exis sequences { n }, n, nd { z n } Ì B such h S n z n y Howeve, he mpping S is coninuous Theefoe, S + n z n = S S n z n S y, n Lemm implies h S y Î w( B) Thus, S w( B) Ì w( B), > 0 Le us pove he evese inclusion Le y Î w( B) Then hee exis sequences { v n } Ì B nd { n : n } such h S n v n y Le us conside he sequence y n = S n - v n, n ³ The sympoic compcness implies h hee exis subsequence nk nd n elemen z Î X such h As sed bove, his gives us h z = lim S nk - y n k k Theefoe, z Î w( B) Moeove, ( ) z = lim S nk - k S z = lim S k S nk - v n = lim S k nk v nk = y k Hence, y ÎS w( B) Thus, he invince of he se w( B) is poved Le us pove he compcness of he se w( B) Assume h { z n } is sequence in w( B) Then Lemm implies h fo ny n we cn find n ³ n nd y n Î B such h z n - S n y n n As sid bove, he popey of sympoic compcness enbles us o find n elemen z nd sequence { n k } such h S nk y nk - z 0, k k y nk
30 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e This implies h z Î w( B) nd z nk z This mens h w( B) is closed nd compc se in H Lemm 5 is poved compleely Now we esblish Theoem 5 Le B be bounded bsobing se of he dynmicl sysem Le us pove h w( B) is globl co I is sufficien o veify h w( B) unifomly cs he bsobing se B Assume he cony Then he vlue sup { dis ( S y, w( B) ): y Î B} does no end o zeo s This mens h hee exis d > 0 nd sequence { n : n } such h ì ü sup ídis ( S n y, w( B) ): y Î Bý ³ d þ Theefoe, hee exiss n elemen y n Î B such h dis ( S n y n, w( B) ) ³ d, n =,, ¼ (5) U As befoe, convegen subsequence { S nk y nk } cn be exced fom he sequence { S n y n } Theewih Lemm implies z º lim S nk y nk k Î w( B) which condics esime (5) Thus, w( B) is globl co Is compcness follows fom he esily veifible elion A º w( B) = Ç Ç ( ) S B > 0 ³ Le us pove he connecedness of he co by educio d bsudum Assume h he co A is no conneced se Then hee exiss pi of open ses U nd such h Ui Ç A ¹ Æ, i =,, A Ì U È U, U Ç U = Æ Le A c = conv ( A) be convex hull of he se A, ie A c = N N ì ü íå l i v i : v i Î A, l i ³ 0, å l i =, N =,, ¼ý i = i = þ I is cle h A c is bounded conneced se nd A c É A The coninuiy of he mpping S implies h he se S A c is lso conneced Theewih A= S A Ì S A c Theefoe, U i Ç S A c ¹ Æ, i =, Hence, fo ny > 0 he pi U, U cnno cove S A c I follows h hee exiss sequence of poins x n = S n y n Î S n A c such h x n Ï U È U The sympoic compcness of he dynmicl sysem enbles us o exc subsequence { n k } such h x nk = S nk y nk ends in X o n elemen y s k I is cle h y Ï U È U nd y Î w( A c ) These equions condic one nohe since w( A c ) Ì w( B) = A Ì U È U Theefoe, Theoem 5 is poved compleely
Theoems on Exisence of Globl Aco 3 I should be noed h he connecedness of he globl co cn lso be poved wihou using he line sucue of he phse spce (do i youself) Execise 5 Show h he ssumpion of sympoic compcness in Theoem 5 cn be eplced by he Ldyzhensky ssumpion: he sequence { S n u n } conins convegen subsequence fo ny bounded sequence { u n } Ì X nd fo ny incesing sequence { n } Ì T + such h n + Moeove, he Ldyzhensky ssumpion is equivlen o he condiion of sympoic compcness Execise 5 Assume h dynmicl sysem ( X, S ) possesses compc globl co A Le A * be miniml closed se wih he popey lim dis ( S y, A * ) = 0 fo evey y Î X Then A * Ì A nd A * = È { w( x): x Î X }, ie A * coincides wih he globl miniml co (cf Execise 34) Execise 53 Assume h equion (47) holds Pove h he globl co A possesses he popey A = w( K) Ì K Execise 54 Assume h dissipive dynmicl sysem possesses globl co A Show h A = w ( B) fo ny bounded bsobing se B of he sysem The fc h he globl co A hs he fom A = w( B), whee B is n bsobing se of he sysem, enbles us o se h he se S B no only ends o he co A, bu is lso unifomly disibued ove i s Nmely, he following sseion holds Theoem 5 Assume h dissipive dynmicl sysem ( X, S ) possesses com- pc globl co A Le B be bounded bsobing se fo ( X, S ) Then lim sup { dis (, S B): Î A} = 0 (5) Poof Assume h equion (5) does no hold Then hee exis sequences { n }Ì Ì A nd { n : n } such h dis ( n, S n B) ³ d fo some d > 0 (53) The compcness of A enbles us o suppose h { n } conveges o n elemen Î A Theewih (see Execise 54) = lim S m y m, { y m } Ì B, m
3 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e whee { m } is sequence such h m Le us choose subsequence { m n } such h mn ³ n + B fo evey n =,, ¼ Hee B is chosen such h S B Ì Ì B fo ll ³ B Le z n = S mn - y Then i is cle h nd n mn { z n } Ì B = lim S mn y mn = lim S n z n n n Equion (53) implies h dis( n, S n z n ) ³ dis( n, S n B) ³ d This condics he pevious equion Theoem 5 is poved Fo descipion of convegence of he jecoies o he globl co i is convenien o use he Husdoff meic h is defined on subses of he phse spce by he fomul ( C, D) = mx{ h( C, D) ; h( D, C) }, (54) whee C, D Î X nd h( C, D) = sup { dis ( c, D): c Î C} (55) Theoems 5 nd 5 give us he following sseion Coolly 5 Le ( X, S ) be n sympoiclly compc dissipive sysem Then is globl co A possesses he popey lim ( S B, A) = 0 fo ny bounded bsobing se B of he sysem ( X, S ) In picul, his coolly mens h fo ny e > 0 hee exiss e > 0 such h fo evey > e he se S B ges ino he e -viciniy of he globl co A; nd vice ves, he co A lies in he e -viciniy of he se S B Hee B is bounded bsobing se The following heoem shows h in some cses we cn ge id of he equiemen of sympoic compcness if we use he noion of he globl wek co Theoem 53 Le he phse spce H of dynmicl sysem ( H, S ) be sepble Hilbe spce Assume h he sysem ( H, S ) is dissipive nd is evolu- iony opeo S is wekly closed, ie fo ll > 0 he wek convegence y n y nd S y n z imply h z = S y Then he dynmicl sysem ( H, S ) possesses globl wek co The poof of his heoem bsiclly epes he esonings used in he poof of Theoem 5 The wek compcness of bounded ses in sepble Hilbe spce plys he min ole insed of he sympoic compcness
Theoems on Exisence of Globl Aco 33 Lemm 5 Assume h he hypoheses of Theoem 53 hold Fo B Ì H he wek w-limi se w w ( B) by he fomul Ç we define w w ( B) = S ( B), s ³ 0 ³È s w (56) whee [ Y] w is he wek closue of he se Y Then fo ny bounded se B Ì H he se w w ( B) is nonempy wekly closed bounded invin se Poof s The dissipiviy implies h ech of he ses g w ( B) = [ È S ( B) ] is ³ s w bounded nd heefoe wekly compc Then he Cno heoem on he collecion of nesed compc ses gives us h w w s ( B) = Ç g w ( B) is nonempy wekly closed bounded se Le us pove is invince Le y Î w w s ³ 0 ( B) Then hee exiss sequence y Î n È S ( B) such h y y wekly The ³ n n dissipiviy popey implies h he se { S y n } is bounded when is lge enough Theefoe, hee exis subsequence { y nk } nd n elemen z such h y nk y nd S y nk z wekly The wek closedness of S implies h s s z = S y Since S y nk Î g w ( B) fo n k ³ s, we hve h z Î g w ( B) fo ll s Hence, z Î w w ( B) Theefoe, S w w ( B) Ì w w ( B) The poof of he evese inclusion is lef o he ede s n execise Fo he poof of Theoem 53 i is sufficien o show h he se A w = w w ( B), (57) whee B is bounded bsobing se of he sysem ( H, S ), is globl wek co fo he sysem To do h i is sufficien o veify h he se B is unifomly ced o A w = w w ( B) in he wek opology of he spce H Assume he cony Then hee exis wek viciniy O of he se A w nd sequences { y n } Ì B nd { n : n } such h S n y n Ï O Howeve, he se { S n y n } is wekly compc Theefoe, hee exis n elemen z Ï O nd sequence { n k } such h z = w - lim S nk y nk k s s Howeve, S nk y nk Î g w ( B) fo nk ³ s Thus, z Î g w ( B) fo ll s ³ 0 nd z Î Î w w ( B), which is impossible Theoem 53 is poved Execise 55 Assume h he hypoheses of Theoem 53 hold Show h he globl wek co A w is conneced se in he wek opology of he phse spce H =È { Execise 56 Show h he globl wek miniml co A* w w w ( x): x Î H} is sicly invin se
34 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e Execise 57 Pove he exisence nd descibe he sucue of globl nd globl miniml cos fo he dynmicl sysem geneed by he equions fo evey el m ì í x = m x - y - x( x + y ), y = x + m y - y( x + y ) Execise 58 Assume h X is meic spce nd ( X, S ) is n sympoiclly compc (in he sense of he definiion given in Remk 4) dynmicl sysem Assume lso h he cing compc K is conined in some bounded conneced se Pove he vlidiy of he sseions of Theoem 5 in his cse In conclusion o his secion, we give one moe sseion on he exisence of he globl co in he fom of execises This sseion uses he noion of he sympoic smoohness (see [3] nd [9]) The dynmicl sysem ( X, S ) is sid o be sympo- iclly smooh if fo ny bounded posiively invin ( S B Ì B, ³ 0) se B Ì X hee exiss compc K such h h( S B, K) 0 s, whee he vlue h (, ) is defined by fomul (55) Execise 59 Pove h evey sympoiclly compc sysem is sympoiclly smooh Execise 50 Le ( X, S ) be n sympoiclly smooh dynmicl sysem Assume h fo ny bounded se B Ì X he se g + B = = È ³ 0 S ( B) is bounded Show h he sysem ( X, S ) possesses globl co A of he fom A = È { w( B): B Ì X, B is bounded } Execise 5 In ddiion o he ssumpions of Execise 50 ssume h ( X, S ) is poinwise dissipive, ie hee exiss bounded se B 0 Ì X such h dis X ( S y, B 0 ) 0 s fo evey poin y Î X Pove h he globl co A is compc 6 On he Sucue of Globl Aco The sudy of he sucue of globl co of dynmicl sysem is n impon poblem fom he poin of view of pplicions Thee e no univesl ppoches o his poblem Even in finie-dimensionl cses he co cn be of compliced sucue Howeve, some ses h undoubedly belong o he co cn be poin-
On he Sucue of Globl Aco 35 be- ed ou I should be fis noed h evey siony poin of he semigoup S longs o he co of he sysem We lso hve he following sseion Lemm 6 Assume h n elemen z lies in he globl co A of dynmicl sysem ( X, S ) Then he poin z belongs o some jecoy g h lies in A wholly Poof Since S A = A nd z Î A, hen hee exiss sequence { z n } Ì A such h z 0 = z, S z n = z n -, n =,, ¼ Theewih fo discee ime he equied jecoy is g = { u n : n ÎZ}, whee u n = S n z fo n ³ 0 nd u n = = z -n fo n 0 Fo coninuous ime le us conside he vlue u() ì = í S z, ³ 0, S + n z n, - n - n +, n =,, ¼ Then i is cle h u() Î A fo ll Î R nd S u() = u( + ) fo ³ 0, Î R Theewih u ( 0) = z Thus, he equied jecoy is lso buil in he coninuous cse Execise 6 Show h n elemen z belongs o globl co if nd only if hee exiss bounded jecoy g = { u(): - < < } such h u ( 0) = z Unsble ses lso belong o he globl co Le Y be subse of he phse spce X of he dynmicl sysem ( X, S ) Then he unsble se emning fom Y is defined s he se M + ( Y) of poins z Î X fo evey of which hee exiss jecoy g = { u(): Î T} such h u ( 0) = z, lim dis ( u(), Y) = 0 - Execise 6 Pove h M + ( Y) is invin, ie S M + ( Y) = M + ( Y) fo ll > 0 Lemm 6 Le N be se of siony poins of he dynmicl sysem ( X, S ) possessing globl co A Then M + ( N ) Ì A Poof I is obvious h he se N = { z : S z = z, > 0} lies in he co of he sysem nd hus i is bounded Le z Î M + ( N ) Then hee exiss jecoy g z = { u(), ÎT} such h u ( 0) = z nd dis ( u( ), N ) 0, -
36 Bsic Conceps of he Theoy of Infinie-Dimensionl Dynmicl Sysems C h p e Theefoe, he se B s = { u( ): -s} is bounded when s > 0 is lge enough Hence, he se S B s ends o he co of he sysem s + Howeve, z Î S B s fo ³ s Theefoe, dis ( z, A) sup { dis ( S y, A): y Î B s } 0, + This implies h z Î A The lemm is poved Execise 63 Assume h he se N of siony poins is finie Show h l M + ( N ) =, È M + ( z k ) k = whee z k e he siony poins of S (he se M + ( z k ) is clled n unsble mnifold emning fom he siony poin z k ) Thus, he globl co A includes he unsble se M + ( N ) I uns ou h unde cein condiions he co includes nohing else We give he following definiion Le Y be posiively invin se of semigoup S : S Y Ì Y, > 0 The coninuous funcionl ( y) defined on Y is clled he Lypunov funcion of he dynmicl sysem ( X, S ) on Y if he following condiions hold: ) fo ny y ÎY he funcion ( S y) is nonincesing funcion wih espec o ³ 0 ; b) if fo some 0 > 0 nd y ÎX he equion ( y) = ( S 0 y) holds, hen y = S y fo ll ³ 0, ie y is siony poin of he semigoup S Theoem 6 Le dynmicl sysem ( X, S ) possess compc co A Assume lso h he Lypunov funcion ( y) exiss on A Then A = M + ( N ), whee N is he se of siony poins of he dynmicl sysem Poof Le y ÎA Le us conside jecoy g pssing hough y (is exisence follows fom Lemm 6) Le g = { u(): Î T} nd g = { u(): } Since g Ì A, he closue [ g ] is compc se in X This implies h he -limi se g ( ) = [ g ] <Ç 0 of he jecoy g is nonempy I is esy o veify h he se ( g) is invin: S ( g) = ( g) Le us show h he Lypunov funcion ( y) is consn on ( g) Indeed, if u Î g ( ), hen hee exiss sequence { n } ending o - such h
On he Sucue of Globl Aco 37 lim u( n ) = u n - Consequenly, ( u) = lim ( u( n )) n By viue of monooniciy of he funcion ( u) long he jecoy we hve ( u) = sup { ( u ( ) ): < 0} Theefoe, he funcion ( u) is consn on ( g) Hence, he invince of he se ( g) gives us h ( S u) = ( u), > 0 fo ll u Îg ( ) This mens h ( g) lies in he se N of siony poins Theewih (veify i youself) lim dis( u (), g ( )) = 0 - Hence, y Î M + ( N ) Theoem 6 is poved Execise 64 Assume h he hypoheses of Theoem 6 hold Then fo ny elemen y ÎA is w-limi se w( y) consiss of siony poins of he sysem Thus, he globl co coincides wih he se of ll full jecoies connecing he siony poins Execise 65 Assume h he sysem ( X, S ) possesses compc globl co nd hee exiss Lypunov funcion on X Assume h he Lypunov funcion is bounded below Show h ny semijecoy of he sysem ends o he se N of siony poins of he sysem s +, ie he globl miniml co coincides wih he se N In picul, his execise confims he fc elized by mny invesigos h he globl co is oo wide objec fo descipion of cully obseved limi egimes of dynmicl sysem Execise 66 Assume h ( R, S ) is dynmicl sysem geneed by he logisic equion (see Exmple ): x + x( x-) = 0, > 0 Show h V( x) = x 3 3 - x is Lypunov funcion fo his sysem Execise 67 Show h he ol enegy E( x, x ) = x + x 4 4 - x - bx is Lypunov funcion fo he dynmicl sysem geneed (see Exmple ) by he Duffing equion x + e x + x 3 - x = b, e > 0