On the Structure of the Global Attractor for Non-Autonomous Dynamical Systems with Weak Convergence

Manusc ip submi ed o Websi e: h p://AIMsciences.o g AIMS’ Jou nals Volume X, Numbe 0X, XX 200X pp. X–XX ON THE STRUCTURE OF THE GLOBAL ATTRACTOR FOR NON-AUTONOMOUS DYNAMICAL SYSTEMS WITH WEAK CONVERGENCE Tom´ as Ca aballo Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico Uni e sidad de Se illa Apdo. de Co eos 1160 41080-Se illa (Spain) Da id Cheban S a e Uni e si y o Moldo a Depa men o Ma hema ics and In o ma ics A. Ma ee ich S ee 60 MD–2009 Chi¸sin˘au, Moldo a Abs ac . The aim o his pape is o desc ibe he s uc u e o global a ac- o s o non-au onomous dynamical sys ems wi h ecu en coe icien s (wi h bo h con inuous and disc e e ime). We conside a special class o his ype o sys ems ( he so–called weak con e gen sys ems). I is shown ha , o weak con e gen sys ems, he answe o Sei e ’s ques ion (Does an almos pe iodic dissipa i e equa ion possess an almos pe iodic solu ion?) is a i ma i e, al- hough, in gene al, e en o scala equa ions, he esponse is nega i e. We s udy his p oblem in he amewo k o gene al non-au onomous dynamical sys ems (cocycles). We apply he gene al esul s ob ained in ou pape o he s udy o almos pe iodic (almos au omo phic, ecu en , pseudo ecu - en ) and asymp o ically almos pe iodic (asymp o ically almos au omo phic, asymp o ically ecu en , asymp o ically pseudo ecu en ) solu ions o di e - en classes o di e en ial equa ions. 1. In oduc ion. Deno e by Rn he n–dimensional eal Euclidian space wi h he no m |·|, and by C(R×Rn,Rn) he space o all con inuous unc ions :R×Rn→Rn equipped wi h he compac -open opology. Conside he di e en ial equa ion x′= ( , x),(1) whe e ∈C(R×Rn,Rn). Assume ha he igh -hand side o (1) sa is ies hypo he- ses ensu ing he exis ence, uniqueness and ex endabili y o solu ions o (1), i.e., o all ( 0, x0)∈R×Rn he e exis s a unique solu ion x( ; 0, x0) o equa ion (1) wi h ini ial da a 0, x0, and de ined o all ≥ 0. Recall (see, o example, [18, 32]) ha equa ion (1) is said o be uni o mly dissi- pa i e (o uni o mly ul ima ely bounded) i he e exis s a numbe 0>0 such ha , 2000 Ma hema ics Subjec Classi ica ion. p ima y:34C11,34C27,34D05,34D23,34D45,34K14, 37B20,37B55,37C55, 7C60, 37C65,37C70,37C75. Key wo ds and ph ases. Non-au onomous dynamical sys ems; skew-p oduc sys ems; cocycles; global a ac o ; dissipa i e sys ems; con e gen sys ems; quasi-pe iodic, almos pe iodic, almos au omo phic, ecu en solu ions; asymp o ically almos pe iodic solu ions. 1 2 TOM´ AS CARABALLO AND DAVID CHEBAN o e e y > 0, he e is L( )>0 such ha , i |x0| ≤ , hen |x( ; 0, x0)| ≤ 0, o all ≥ 0+L( ). Equa ion (1) ( espec i ely, he unc ion ) is called egula , i o e e y x0∈Rn and g∈H( ) := { τ:τ∈R}(whe e by ba we deno e he closu e in he space C(R×Rn,Rn) and τ( , x) := ( +τ, x) o all ( , x)∈R×Rn), he equa ion x′=g( , x) (2) possesses a unique solu ion ϕ( , x, g) passing h ough he poin x0a he ini ial momen = 0, and de ined on R+:= { ∈R| ≥0}. Theo em 1.1. [9, Ch.II] Suppose ha ∈C(R×Rn,Rn)and H( )is a compac subse o C(R×Rn,Rn). Then, he ollowing s a emen s a e equi alen : (i) equa ion (1) is uni o mly dissipa i e; (ii) he e exis s a posi i e numbe R0such ha lim sup →+∞|ϕ( , x, g)| ≤ R0,(3) o all (x, g)∈Rn×H( ). A ligh o Theo em 1.1, i is said ha equa ion (1) is dissipa i e (in ac , he am- ily o equa ions (2) is collec i ely dissipa i e, bu we use his sho e e minology) i (3) holds. Sei e ’s P oblem (see [19] o mo e de ails): Suppose ha equa ion (1) is dissipa i e and he unc ion is almos pe iodic (wi h espec o he ime a iable). Does equa ion (1) possess an almos pe iodic solu ion? Fink and F ede icson [19] and Zhiko [33] es ablished ha , in gene al, e en when equa ion (1) is scala , he answe o Sei e ’s ques ion is nega i e. Rela ed o his esul , he e a e he ollowing in e es ing ques ions: a) To ex ac some classes o dissipa i e di e en ial equa ions o which he e- sponse o Sei e ’s p oblem is posi i e; b) To indica e he addi ional (i is desi able “op imal”) condi ions which, join ly wi h dissipa i i y, gua an ee he exis ence o a leas one almos pe iodic solu ion o equa ion (1). Below we include a sho su ey on esul s conce ning he ques ions a) and b). a) Fo he ollowing classes o dissipa i e equa ions o ype (1), he esponse o Sei e ’s ques ion is a i ma i e: linea equa ions (see [9, Ch.II]), quasi-linea equa- ions (weak non-linea pe u ba ions o linea equa ions) (see [7, 9]); holomo phic equa ions (see [5, 6, 8, 9]). b) Zubo (see [36]) es ablished ha equa ion (1) admi s a unique almos pe iodic solu ion i i is con e gen , i.e., i admi s a unique solu ion which is bounded on Rand also uni o mly globally asymp o ically s able. This esul was gene alized o equa ions (1) wi h ecu en coe icien s by Cheban [9, Ch.II] and wi h pseudo ecu en coe icien s by Ca aballo and Cheban [4]. Le i an and Zhiko (see, o example, [23]) p o ed ha , o low-dimensional equa ions (namely, o n≤3), equa ion (1) admi s a leas one almos pe iodic solu ion i (1) is uni o mly posi i ely s able, i.e., o all ε > 0 he e exis s δ= δ(ε)>0 such ha |x1−x2|< δ implies |ϕ( , x1, g)−ϕ( , x2, g)|< ε, o all ≥0 and g∈H( ). The main esul o ODEs (Theo em 4.2 and i s gene aliza ions) ha we p o e in his pape is he ollowing: we show ha i equa ion (1) is weak con e gen (i.e., ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 3 he e exis s a posi i e numbe Lsuch ha lim →+∞|ϕ( , x1, g)−ϕ( , x2, g)|= 0 o all |xi| ≤ L(i= 1,2) and g∈H( )), and is pseudo ecu en wi h espec o he ime a iable (in pa icula , is ecu en , almos au omo phic, Boh almos pe iodic o quasi pe iodic), hen, equa ion (1) admi s a unique pseudo ecu en ( espec i ely, ecu en , almos au omo phic, Boh almos pe iodic, quasi pe iodic) solu ion. I his solu ion is Lyapuno s able, hen he Le inson cen e ( he compac global a ac o ) is a minimal almos pe iodic se . I i is no Lyapuno s able, hen he Le inson cen e con ains a minimal almos pe iodic se , bu i is no minimal ( his means, in pa icula , ha equa ion (1) admi s a amily (mo e han one) o solu ions which a e bounded on R). We p esen ou esul s in he amewo k o gene al non-au onomous dynamical sys ems (cocycles) and we apply ou abs ac heo y o se e al classes o di e en ial equa ions. The pape is o ganized as ollows. In Sec ion 2, we collec some no ions (global a ac o , minimal se , poin /compac dissipa i i y, non-au onomous dynamical sys ems wi h con e gence, quasi pe iodic- i y, Le i an/Boh almos pe iodici y, almos au omo phy, ecu ence, pseudo ecu - ence, Poisson s abili y, e c) and ac s om he heo y o dynamical sys ems which will be necessa y in his pape . Sec ion 3 is de o ed o he s udy o a special class o non-au onomous dynamical sys ems (NDS): he so-called NDS wi h weak con e gence. We gi e a gene aliza ion o he no ion o con e gen NDS. On he one hand, his ype o NDS is e y close o NDS wi h con e gence (because hey conse e some p ope ies o con e gen sys- ems) and la ge han ha o con e gen sys ems. On he o he hand, we analyze he class o compac dissipa i e NDS wi h non i ial Le inson cen e . The main esul s o ou pape a e p o ed in his sec ion, namely Theo em 3.5 and Theo em 3.8 (see also Co olla y 3.9 and Co olla y 3.10) which p o ide su icien condi ions o he exis ence o a unique minimal se in he Le inson cen e which is homeo- mo phic o he base dynamical sys em (d i ing sys em). This means, in pa icula , ha i he base dynamical sys em is a compac minimal se consis ing o ecu en ( espec i ely, almos au omo phic, Boh almos pe iodic, quasi pe iodic, pe iodic, s a iona y) poin s, hen, unde he condi ions o Theo em 3.5, he Le inson cen- e o a non-au onomous dynamical sys em con ains a unique minimal se which consis s o ecu en ( espec i ely, almos au omo phic, Boh almos pe iodic, quasi pe iodic, pe iodic, s a iona y) poin s. In Sec ion 4, we exhibi some applica ions o ou abs ac esul s o di e en classes o di e en ial equa ions. Namely, almos pe iodic and asymp o ically al- mos pe iodic solu ions (Subsec ion 4.1), uni o mly compa ible (by he cha ac e o ecu ence wi h he igh hand side) solu ions o s ic dissipa i e equa ions (Sub- sec ion 4.2). 2. Nonau onomous Dynamical Sys ems wi h Con e gence. Le us s a by ecalling some concep s and no a ions abou he heo y o non-au onomous dynam- ical sys ems which will be necessa y o ou analysis. 2.1. Compac Global A ac o s o Dynamical Sys ems. Le (X, ρ) be a me ic space, R(Z) be he g oup o eal (in ege ) numbe s, R+(Z+) be he semi- g oup o nonnega i e eal (in ege ) numbe s, Sbe one o he wo se s Ro Zand T⊆S(S+⊆T) be a sub-semig oup o he addi i e g oup S. 4 TOM´ AS CARABALLO AND DAVID CHEBAN Adynamical sys em is a iple (X, T, π), whe e π:T×X→Xis a con inuous mapping sa is ying he ollowing condi ions: π(0, x) = x(∀x∈X); π(s, π( , x)) = π(s+ , x) (∀ , s ∈Tand x∈X). I T=R(R+) o Z(Z+), he dynamical sys em (X, T, π) is called a g oup (semi- g oup). When T=R+o R, he dynamical sys em (X, T, π) is called a low, bu i T⊆Z, hen (X, T, π) is called a cascade (disc e e low). The unc ion π(·, x) : T→Xis called he mo ion passing h ough he poin xa he ini ial momen = 0, and he se Σx:= π(T, x) is called he ajec o y o his mo ion. A nonemp y se M⊆Xis called posi i ely in a ian (nega i ely in a ian ,in- a ian ) wi h espec o he dynamical sys em (X, T, π) o , simply, posi i ely in a i- an (nega i ely in a ian , in a ian ), i π( , M)⊆M(M⊆π( , M), π( , M) = M) o e e y ∈T. A closed posi i ely in a ian se is said o be minimal i i does no con ain any own closed posi i ely in a ian subse . I is easy o see ha e e y posi i ely in a ian minimal se is in a ian . Le M⊆X. The se ω(M) := ≥0[ τ≥ π(τ, M) is called he ω-limi o M. The se Ws(Λ), de ined by he equali y Ws(Λ) := {x∈X|lim →+∞ρ(π( , x),Λ) = 0} is called he s able mani old o he se Λ ⊆X. Fo p∈X,M⊂Xand δ > 0, le us deno e by B(M, δ) = {x∈X|ρ(x, M)< δ}. The se Mis called: –o bi ally s able i o e e y ε > 0, he e exis s δ=δ(ε)>0 such ha ρ(x, M)< δimplies ρ(π( , x), M)< ε, o all ≥0; –a ac ing i he e exis s γ > 0 such ha B(M, γ)⊂Ws(M); –asymp o ically s able i i is o bi ally s able and a ac ing; –globally asymp o ically s able i i is asymp o ically s able and Ws(M) = X. The dynamical sys em (X, T, π) is called: −poin dissipa i e i he e exis s a nonemp y compac subse K⊆Xsuch ha , o e e y x∈X, lim →+∞ρ(π( , x), K) = 0; (4) −compac dissipa i e i he equali y (4) akes place uni o mly w. . . xon he compac subse s o X; −locally comple e (compac ) i o any poin p∈X, he e exis δp>0 and lp>0 such ha he se π(lp, B(p, δp)) is ela i ely compac . Le (X, T, π) be compac dissipa i e, and Kbe a compac se a ac ing e e y compac subse o X. Le us se J:= ω(K) := ≥0[ τ≥ π(τ, K).(5) I can be shown (see [9, Ch.I]) ha he se Jde ined by equali y (5) does no depend on he choice o he a ac o K, bu is cha ac e ized only by he p ope ies ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 5 o he dynamical sys em (X, T, π) i sel . The se Jis called he Le inson cen e o he compac dissipa i e dynamical sys em (X, T, π). Some p ope ies o his se can be ound in [9, 20]. 2.2. Non-Au onomous Dynamical Sys ems wi h Con e gence. Gi en wo dynamical sys ems (X, T1, π) and (Y, T2, σ), a iple h(X, T1, π),(Y, T2, σ), hi, whe e his a homomo phism om (X, T1, π) on o (Y, T2, σ), is called a non-au onomous dynamical sys em (see [2, 9]). Recall (see, o example, [9, 12]) ha he non- au onomous dynamical sys em h(X,T1, π),(Y, T2, σ), hiis said o be con e gen i he ollowing condi ions a e sa is ied: (i) he dynamical sys ems (X, T1, π) and (Y, T2, σ) a e compac dissipa i e; (ii) he se JXTXycon ains no mo e han one poin o all y∈JY, whe e Xy:= h−1(y) := {x∈X|h(x) = y}and JX( espec i ely, JY) is he Le inson cen e o he dynamical sys em (X, T1, π) ( espec i ely, (Y, T2, σ)). Some su icien condi ions and c i e ia ensu ing he con e gence o a dynamical sys em can be ound in [9, Ch.II]. Thus, a non-au onomous dynamical sys em h(X, T1, π),(Y, T2, σ), hiis con e - gen , i he sys ems (X, T1, π) and (Y, T2, σ) a e compac dissipa i e wi h Le inson cen e s JXand JY espec i ely, and JXhas “ i ial” sec ions, i.e., JXTXyconsis s o a single poin o all y∈JY. In his case, he Le inson cen e JXo he dynam- ical sys em (X, T1, π) is a copy (a homeomo phic image) o he Le inson cen e JY o he dynamical sys em (Y, T2, σ). Thus, he dynamics on JXis he same as on JY. Rema k 2.1. 1. We no e ha con e gen sys ems a e, in some sense, he simples dissipa i e dynamical sys ems. I Yis compac and in a ian , T2=S,h(X, T1, π), (Y, T2, σ), hiis a con e gen non-au onomous dynamical sys em, and JXis he Le inson cen e o (X, T1, π), hen (JX,T2, π) and (Y, T2, σ) a e homeomo phic. Al hough he Le inson cen e o a con e gen sys em can be comple ely desc ibed, i may be su icien ly complica ed. 2. The concep o con e gen sys em o di e en ial equa ions is well de eloped (see, o example, B. P. Demido ich [16, 17, 18], Pa lo e al. [24], V. A. Pliss [25, 26], V. I. Zubo [35], and many o he s). The non-au onomous sys em o di e en ial equa ions x′= ( , x) (6) is called con e gen , i i admi s a unique solu ion de ined and bounded on R, which is uni o mly globally asymp o ically s able. I is possible o show ha he non-au onomous dynamical sys em gene a ed by he con e gen equa ion (6) is con e gen . Howe e , he concep o con e gen non-au onomous dynamical sys em is much mo e gene al (see [9, 12] and he bibliog aphy he ein). Gi en ε > 0,a numbe τ∈Tis called an ε−shi ( espec i ely, an ε−almos pe iod) o x, i ρ(π(τ, x), x)< ε ( espec i ely, ρ(π(τ+ , x), π( , x)) < ε, o all ∈T). A poin x∈Xis called almos ecu en ( espec i ely, Boh almos pe iodic), i o any ε > 0, he e exis s a posi i e numbe lsuch ha , in any segmen o leng h l, he e is an ε−shi ( espec i ely, an ε−almos pe iod) o he poin x∈X. I he poin x∈Xis almos ecu en , and he se H(x) := {π( , x)| ∈T}is compac , hen xis called ecu en , whe e he ba deno es he closu e in X. 6 TOM´ AS CARABALLO AND DAVID CHEBAN Deno e by Nx:= {{ n} ⊂ T: such ha {π( n, x)} → xand { n} → ∞} and Mx:= {{ n} ⊂ T: such ha {π( n, x)}is con e gen and { n} → ∞}. A poin x∈Xis called Poisson s able in he posi i e di ec ion i he e exis s a sequence { n} ∈ Nxsuch ha n→+∞as n→ ∞. Le (X, T, π) be a wo-sided dynamical sys em (i.e., T=S). A poin x∈Xis called Poisson s able in he nega i e di ec ion i he e exis s a sequence { n} ∈ Nx such ha n→ −∞ as n→ ∞. The poin x∈Xis called Poisson s able i i is Poisson s able in bo h di ec ions. The dynamical sys em (X, T, π) is said o be (i) ansi i e i he e exis s a poin x0∈Xsuch ha H(x0) = X; (ii) pseudo ecu en i Xis compac , ansi i e, and e e y poin x∈Xis Poisson s able. A poin x∈Xis called pseudo ecu en (see [28, 30]) i he dynamical sys em (H(x),T, π) is pseudo ecu en . Rema k 2.2. E e y ecu en poin is pseudo ecu en , bu he e exis pseudo ecu en poin s which a e no ecu en (see [28, 30]). An m-dimensional o us is deno ed by Tm:= Rm/2πZm.Le (Tm,T, σ) be an i a ional winding o Tm, i.e., σ( , ν) := (ν1 , ν2 ,...,νm ) o all ∈Sand ν∈ Tm. A poin x∈Xis called quasi-pe iodic, wi h equency ν:= (ν1, ν2,...,νm)∈ Tm, i he e exis s a con inuous unc ion Φ : Tm→Xsuch ha π( , x) := Φ(σ( , ω)) o all ∈T,whe e (Tm,T, σ) is an i a ional winding o he o us Tmand ω∈ Tm. A poin x∈Xo he dynamical sys em (X, T, π) is called Le i an almos pe iodic (see [2, 23]) i he e exis s a dynamical sys em (Y, T, σ), and a Boh almos pe iodic poin y∈Ysuch ha Ny⊆Nx. Rema k 2.3. Le xi∈Xi(i= 1,2,...,m) be a Le i an almos pe iodic poin o he dynamical sys em (Xi,T, πi).Then, he poin x:= (x1, x2,...,xm)∈X:= X1×X2×... ×Xmis also Le i an almos pe iodic in he p oduc dynamical sys em (X, T, π),whe e π:T×X→Xis de ined by he equali y π( , x) := (π1( , x1), π2( , x2), . . . , πm( , xm)), o all ∈Tand x:= (x1, x2,...,xm)∈X. Recall (see [11]) ha he poin x∈Xis called asymp o ically τ–pe iodic ( espec- i ely, asymp o ically quasi pe iodic,asymp o ically Boh almos pe iodic,asymp- o ically almos au omo phic,asymp o ically ecu en ,asymp o ically pseudo ecu - en ) i he e exis s a τ-pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, almos au omo phic, ecu en , pseudo ecu en ) poin p∈Xsuch ha lim →+∞ρ(π( , x), π( , p)) = 0. Thus, we can p o e he ollowing esul . Lemma 2.4. Le h(X, T1, π),(Y, T2, σ), hibe a con e gen non-au onomous dynam- ical sys em, and JX( espec i ely, JY) be he Le inson cen e o he dynamical sys- em (X, T1, π)( espec i ely, (Y, T2, σ)), and y0∈JYbe a τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, ecu en , pseudo ecu en ) poin . Then, he ollowing s a emen s hold: (i) he poin x0∈JXTXy0is also τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, ecu en , pseudo ecu en ); (ii) e e y poin x∈Xy0is asymp o ically τ–pe iodic ( espec i ely, asymp o ically quasi pe iodic, asymp o ically Boh almos pe iodic, asymp o ically ecu en , asymp o ically pseudo ecu en ). ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 7 P oo . The i s s a emen ollows di ec ly om he co esponding de ini ion. Le x∈Xy0be an a bi a y poin . We will show ha lim →+∞ρ(π( , x), π( , x0)) = 0. Indeed, i we suppose ha i is no ue, hen he e a e ε0>0 and n→+∞ ({ n} ⊆ T1) such ha ρ(π( n, x), π( n, x0)) ≥ε0.(7) Since he dynamical sys em (X, T1, π) is compac dissipa i e, we may suppose ha he sequences {π( n, x)}and {π( n, x0)}a e con e gen . Deno e by p:= lim n→∞π( n, x), and p0:= lim n→∞π( n, x0). Then, p, p0∈Xq⊆JX, whe e q:= lim n→∞σ( n, y0). Since he non-au onomous dynamical sys em h(X, T1, π),(Y, T2, σ), hi is con e gen , hen p=p0. On he o he hand, passing o he limi in (7), we ob ain ρ(p, p0)≥ε0>0. This con adic ion p o es ou s a emen . 3. Non-Au onomous Dynamical Sys ems wi h Weak Con e gence. In his sec ion we will s udy a class o non-au onomous dynamical sys ems which is e y close o con e gen sys ems, bu possessing a non- i ial global a ac o . This means ha his class o non-au onomous sys ems will conse e almos all p ope ies o con e gen sys ems, bu will ha e a “non- i ial” global a ac o JX, i.e., he e exis s a leas one poin y∈JYsuch ha he se JXTXycon ains mo e han one poin . A non-au onomous dynamical sys em h(X, T1, π),(Y, T2, σ), hiis said o be weak con e gen , i he ollowing condi ions hold: (i) he dynamical sys ems (X, T1, π) and (Y, T2, σ) a e compac dissipa i e wi h Le inson cen e s JXand JY espec i ely; (ii) i ollows ha lim →+∞ρ(π( , x1), π( , x2)) = 0, o all x1, x2∈JXwi h h(x1) = h(x2). Rema k 3.1. I is clea ha e e y con e gen non-au onomous dynamical sys em is weak con e gen . The opposi e s a emen is no ue in gene al. Indeed, he las s a emen can be con i med by he ollowing example. Le (X, T, π) be an au onomous dynamical sys em wi h compac global a ac o J, which possesses a unique s a iona y a ac ing poin p(i.e., lim | |→+∞ρ(π( , x), p) = 0 o all x∈J) and J6={p}). Fo example, conside he dynamical sys em (X, R, π) on he space X=R2, gene a ed by ollowing sys em o di e en ial equa ions            x′=x2(2y−x) + 2y5 (x2+y2)[1 + (x2+y2)2] y′=8y2(y−x) (x2+y2)[1 + (x2+y2)2]. (8) The phase plane o his dynamical sys em (8) is desc ibed in Figu e 1. Fo mo e de ails see [31] and also [3, Ch.II, pp.94-98] and [21, Ch.V, pp.191-194]. Fo a simila example he eade is e e ed o [1, Ch.V, p.59] (Example 1.7.8). We can now p o e he ollowing esul which will be c ucial in he p oo o one o ou main esul s (namely, Theo em 3.5). Lemma 3.2. Le h(X, T1, π),(Y, T2, σ), hibe a non-au onomous dynamical sys em, and assume ha he ollowing condi ions hold: 8 TOM´ AS CARABALLO AND DAVID CHEBAN Figu e 1. (i) Yis a compac minimal se ; (ii) he e exis s a poin x0∈Xwi h ela i ely compac posi i e semi- ajec o y Σ+ x0={π( , x0)| ≥0}; (iii) lim →+∞ρ(π( , x1), π( , x2)) = 0, o all x1, x2∈Xwi h h(x1) = h(x2). Then, he e exis s a unique compac minimal se M⊆Xsuch ha : (i) he sec ion MTXyo he se Mconsis s o a single poin my o all y∈Y; (ii) e e y posi i e semi- ajec o y Σ+ xis ela i ely compac ; (iii) lim →+∞ρ(π( , x), mσ( ,h(x))) = 0,(9) o all x∈X. P oo . Since he posi i e semi- ajec o y Σ+ x0o x0is ela i ely compac , he ω–limi se ωx0o he poin x0is nonemp y, compac , in a ian and con ains a leas one minimal subse M⊆ωx0. We will p o e ha he dynamical sys em (X, T1, π) has a mos one minimal se . Indeed, i we suppose ha M1and M2a e wo di e en minimal se s o (X, T1, π), hen M1TM2=∅and, in pa icula , M1yTM2y=∅ o all y∈Y, whe e Miy := h−1(y)TMi(i= 1,2). Le xi∈Miy and n→+∞ such ha σ( n, y)→yand π( n, xi)→¯xi∈Miy (i= 1,2) as n→ ∞; lim n→∞ρ(π( n, x1), π( n, x2)) = 0.(10) I is now easy o see ha he e exis s such a sequence. F om he equali y (10) we ha e ¯x1= ¯x2∈M1yTM2y. This is a con adic ion and, he e o e, Mis he unique compac minimal se o he dynamical sys em (X, T1, π). Le y∈Ybe an a bi a y poin , hen i is ecu en . By Lemma 6.5.19 in [12, Ch.VI, p.226], he e exis s a unique ecu en poin my∈Mysuch ha he equali y (9) holds o all x∈My. Now, we will p o e ha My={my}. Indeed, i x∈My hen, he e exis s a sequence n→+∞such ha π( n, my)→xbecause Mis minimal. On he o he hand, σ( n, y)→yand, since he poin myis uni o mly compa ible by he cha ac e o ecu ence wi h he poin y, hen π( n, my)→my. Thus, we ha e x=myand, consequen ly, My={my} o all y∈Y. ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 9 To inish he p oo i is su icien o no e ha lim →+∞ρ(π( , x), mσ( ,h(x))) = 0, o all x∈X. Le (X, T, π) be a dynamical sys em. Deno e by ΩX:= ∪{ωx|x∈X}and by D+(M) := Tε>0S ≥0π( , B(M, ε)), whe e M⊆X. Co olla y 3.3. Unde he assump ions in Lemma 3.2, he dynamical sys em (X, T1, π)is poin dissipa i e and ΩXis a compac minimal se . Below we gi e an example o poin dissipa i e, bu no compac dissipa i e, dynamical sys em wi h weak con e gence. Example 3.4. Le ϕ∈C(R,R) be a unc ion possessing he ollowing p ope ies: 1. ϕ(0) = 0; 2. supp(ϕ) = [0,2]; 3. ϕ∈C∞(R,R); 4. ϕ(1) = 1; 5. he unc ion ϕis mono one inc easing om 0 o 1 and i is dec easing om 1 o 2; 6. xϕ(x−1)→0 as x→+∞. A unc ion ϕwi h p ope ies 1.−6.can be cons uc ed as ollows. Le ϕ0( ) := exp [ 2−1]−1+ 1,| |<1 0,| | ≥ 1. Then, he unc ion ϕ( ) := ϕ0( −1) is as desi ed. We se X:= {aϕ  a+h|h∈ R, a > 0}∪{θ}, whe e θis he unc ion om C(R,R) iden ically equal o 0. I is possible o show ha he se Xis closed in C(R,R), and i is in a ian wi h espec o shi s. Thus, on he se Xis induced a dynamical sys em (on C(R,R) is de ined he dynamical sys em o ansla ions o Bebu o ’s dynamical sys em), which we deno e by (X, R, σ). We will indica e some p ope ies o his sys em: (i) o e e y unc ion ψ∈X he se {σ( , ψ) : ∈R}is ela i ely compac and ωψ=αψ={θ}, whe e αψdeno es he α-limi associa ed o ψ; (ii) he dynamical sys em (X, R, σ) is poin wise dissipa i e, and ΩX={θ}; (iii) D+(ΩX) = Xand, consequen ly, he dynamical sys em (X, R, σ) is no com- pac dissipa i e because he se X, e iden ly, is no compac ; (i ) he dynamical sys em (X, R, σ) does no admi a maximal compac in a ian se . The necessa y example is he e o e cons uc ed. The subse A⊆Xis said o be chain ansi i e (see [15, 22]) i o any a, b ∈A, and any ε > 0 and L > 0, he e a e ini e sequences x1, x2,...,xm∈Awi h a= x1, b =xm, and 1, 2,..., m≥Lsuch ha ρ(π( i, xi), xi+1)< ε (1 ≤i≤m−1). The sequence {x1, x2,...,xm}is called an ε–chain in Aconnec ing aand b. We can now es ablish and p o e he main esul s o ou pape . Theo em 3.5. Le h(X, T1, π),(Y, T2, σ), hibe a non-au onomous dynamical sys- em sa is ying he ollowing condi ions: (i) Yis a compac minimal se ; (ii) he dynamical sys em (X, T1, π)is compac dissipa i e wi h Le inson cen e J; 16 TOM´ AS CARABALLO AND DAVID CHEBAN In his sec ion we suppose ha equa ion (18) is egula . Equa ion (18) is called dissipa i e (see [9]), i he e exis s a posi i e numbe such ha lim sup →+∞|ϕ( , u, y)|< o all u∈Rnand y∈Y, whe e |·| is a no m in Rn. I is well known (see [19, 33]) ha a dissipa i e equa ion wi h almos pe iodic coe icien s (Yis an almos pe iodic minimal se ) does no ha e, in gene al, an almos pe iodic solu ion. Fo ce ain classes o dissipa i e equa ions o he o m (18), in he wo ks [5]–[8] one can ind su icien condi ions o he exis ence o a leas one almos pe iodic solu ion. In his subsec ion we gi e a simple geome ic condi ion which gua an ees exis ence o a unique almos pe iodic solu ion, and his solu ion, in gene al, is no he unique solu ion o equa ion (18) which is bounded on R. We can now es ablish he ollowing in e es ing esul . Theo em 4.2. Suppose ha he ollowing condi ions a e ul illed: (i) equa ion (18) is egula and dissipa i e; (ii) he space Yis compac , and he dynamical sys em (Y, R, σ)is minimal; (iii) o all y∈Y lim →+∞|ϕ( , u1, y)−ϕ( , u2, y)|= 0,(21) whe e ϕ( , ui, y)(i= 1,2) is he solu ion o equa ion (18) passing h ough ui a he ini ial momen = 0, which is bounded on R. Then, (i) i he poin yis τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, almos au omo phic, ecu en ), hen equa ion (18) admi s a unique τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, almos au omo phic, ecu - en ) solu ion ϕ( , uy, y)(uy∈Rn); (ii) e e y solu ion ϕ( , x, y)is asymp o ically τ–pe iodic ( espec i ely, asymp o - ically quasi pe iodic, asymp o ically Boh almos pe iodic, asymp o ically al- mos au omo phic, asymp o ically ecu en ). P oo . Le hRn, ϕ, (Y, R, σ)ibe he cocycle associa ed o equa ion (18). Deno e by (X, R+, π) he skew-p oduc dynamical sys em, whe e X:= Rn×Yand π:= (ϕ, σ) (i.e., π( , (u, y)) := (ϕ( , u, y), σ( , y)) o all x:= (u, y)∈Rn×Yand ∈R+). Conside he non-au onomous dynamical sys em h(X, R+, π),(Y, R, σ), higene a ed by he cocycle ϕ( espec i ely, by equa ion (18)), whe e h:= p 2:X→Yis he p ojec ion wi h espec o he second a iable, i.e., p 2(α, y) = y. Since Yis compac , i is e iden ha he dynamical sys em (Y, R, σ) is compac dissipa i e and i s Le inson cen e JYcoincides wi h Y. By Theo em 2.23 in [9], he skew- p oduc dynamical sys em (X, R+, π) is compac dissipa i e. Deno e by JXi s Le inson cen e and by Iy:= p 1(JXTXy) o all y∈Y, whe e Xy:= {x∈X: h(x) = y}, and p 1is he p ojec ion unc ion wi h espec o he i s a iable, i.e. p 1(α, y) = α. Acco ding o he de ini ion o he se Iy⊆Rn, and by Theo em 2.24 in [9], u∈Iyi and only i he solu ion ϕ( , u, y) is de ined on Rand bounded (i.e., he se ϕ(R, u, y)⊆Rnis compac ). Thus, Iy={u∈Rn|(u, y)∈JX}. I is easy o see he condi ion (21) means ha he non-au onomous dynamical sys em h(X, R+, π),(Y, R, σ), hiis weak con e gen . To inish he p oo , i is su icien o apply Lemma 6.5.19 in [12, Ch.VI, p.226] and Co olla y 3.10 o he non-au onomous sys em h(X, R+, π),(Y, R, σ), higene a ed by equa ion (18). ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 17 Rema k 4.3. Unde he condi ions o Theo em 4.2 he e exis s a unique almos pe iodic solu ion o equa ion (18), bu equa ion (18) has, gene ally speaking, mo e han one solu ion de ined and bounded on R. Below, we will gi e an example which con i ms his s a emen . Example 4.4. Conside he ollowing almos pe iodic sys em o wo di e en ial equa ions        u′=(u−sin )2(2 −u+2 sin √2 −sin )+2( −sin √2 )5 ((u−sin )2+( −sin √2 )2)[1+((u−sin )2+( −sin √2 )2)2]+ cos ′=8( −sin √2 )2( −u+sin √2 −sin ) ((u−sin )2+( −sin √2 )2)[1+((u−sin )2+( −sin √2 )2)2]+√2 cos √2 . (22) I is easy o check ha he almos pe iodic unc ion ϕ:R→R2de ined by he equali y ϕ( ) := (sin , sin √2 ) is a solu ion o sys em (22). Le now x:= u−sin and y:= −sin √2 . Then, he sys em (22) educes o (8). Thus, e e y solu ion φ o he sys em (22) possesses he o m φ=ϕ+ψ, whe e ψis some solu ion o he sys em (8). Since (8) is weak con e gen and admi s mo e han one solu ion which is bounded on R, he sys em (22) possesses he same p ope y. 4.2. Uni o m compa ible solu ions o s ic dissipa i e equa ions. In his sec ion we conside equa ion (18) when he d i ing sys em (Y, R, σ) is pseudo e- cu en , and he unc ion ∈C(Y×Rn,Rn) is s ic dissipa i e wi h espec o i s second a iable x∈Rn, i.e., h (y, x1)− (y, x2), x1−x2i<0 (23) o all x1, x2∈Rn(x16=x2) and y∈Y. Recall (see [28, 29, 30]) ha he poin x∈Xis called compa able ( espec i ely, uni o mly compa able) by he cha ac e o ecu ence wi h he poin y∈Yi Ny⊆ Nx( espec i ely, My⊆Mx). Le us now ecall a esul which plays an impo an ole in he p oo o ou main esul in his subsec ion. Theo em 4.5. (See [28, 30]) Le (X, T, π)and (Y, T, σ)be wo dynamical sys ems, x∈Xand y∈Y. Then, he ollowing s a emen s hold: (i) I xis compa able by he cha ac e o ecu ence wi h y, and yis τ–pe iodic ( espec i ely, Le i an almos pe iodic, almos ecu en , Poisson s able), hen so is he poin x. (ii) I xis uni o mly compa able by he cha ac e o ecu ence wi h y, and yis τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, almos au o- mo phic, ecu en ), hen so is he poin x. Following B. A. Shche bako [28, 30], a solu ion ϕ( , u, y) o equa ion (18) is said o be compa ible ( espec i ely, uni o mly compa ible) by he cha ac e o ecu ence wi h he igh hand-side i Ny⊆Nϕ(·,u,y)( espec i ely, My⊆Mϕ(·,u,y)). Theo em 4.6. Le (Y, R, σ)be pseudo ecu en , ∈C(Y×Rn,Rn)be s ic dissipa i e wi h espec o he a iable x, and assume ha he e exis s a leas one solu ion ϕ( , x0, y)o equa ion (18) which is bounded on R+. Then, (i) equa ion (18) is con e gen , i.e., he cocycle ϕassocia ed o equa ion (18) is con e gen ; 18 TOM´ AS CARABALLO AND DAVID CHEBAN (ii) o all y∈Y, equa ion (18) admi s a unique solu ion ϕ( , xy, y)which is bounded on Rand uni o mly compa ible, i.e., My⊆Mϕ(·,xy,y); (iii) i he poin yis τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, almos au omo phic, ecu en ), hen (a) equa ion (18) has a unique τ–pe iodic ( espec i ely, quasi pe iodic, Boh almos pe iodic, almos au omo phic, ecu en ) solu ion; (b) e e y solu ion ϕ( , x, y)is asymp o ically τ–pe iodic ( espec i ely, asymp- o ically quasi pe iodic, asymp o ically Boh almos pe iodic, asymp o i- cally almos au omo phic, asymp o ically ecu en ); (c) lim →∞|ϕ( , x, y)−ϕ( , xy, y)|= 0 o all x∈Rnand y∈Y. P oo . Le V:X˙ ×X→R+be he mapping de ined by he equali y V((x1, y), (x1, y)) := |x1−x2|2 o all x1, x2∈Rnand y∈Y, whe e | · |2:= h·,·i and hξ1, ξ2i:= n P i=1 ξ1 iξ2 i(ξj:= (ξj 1, ξj 2,...,ξj n)∈Rn(j= 1,2)). Le (X, R+, π) be he skew-p oduc dynamical sys em associa ed o he cocycle ϕ. Then, dV (π( , (u1, y)), π( , (u2, y))) d  =0 =h (y, u1)− (y, u2), u1−u2i<0 (24) o all u1, u2∈Rn(u16=u2) and y∈Y. F om (24) i ollows ha V(π( , (u1, y)), π( (u2, y)) < V ((u1, y),(u2, y)) o all u1, u2∈Rn(u16=u2) and y∈Y. Now o p o e he i s s a emen i is su icien o apply Theo em 3.11. Acco ding o he i s s a emen o he heo em, he skew-p oduc dynamical sys- em (X, R+, π) is compac dissipa i e and, i JXis i s Le inson cen e , hen JXTXy consis s o a single poin xy= (uy, y) and ϕ( , uy, y) is he unique solu ion o equa- ion (18) de ined and bounded on R. Now, we will p o e ha he solu ion ϕ(·, uy, y) is uni o mly compa ible by he cha ac e o ecu ence, i.e., My⊆Mϕ(·,uy,y). I is easy o see ha he las s a emen is equi alen o he inclusion My⊆Mxy. Le { k} ∈ My. Then, he e exis s a poin q∈Ysuch ha σ( n, y)→qas n→ ∞. Conside he sequence {π( k, xy)}. Since xy∈JX, he sequence {π( k, xy)}is ela- i ely compac . Le p1and p2be wo poin s o accumula ion o his sequence. Then, he e exis wo subsequences { (i) k} ⊆ { k}(i= 1,2) such ha pi= lim k→∞π( i k, xy) (i= 1,2). Since JXis a compac in a ian se , hen pi∈JX(i= 1,2). On he o he hand, π( i k, xy) = (ϕ( i k, uy, y), σ( i k, y)) →(¯ui, q) = piand, consequen ly, pi∈Xq. Thus pi∈JXTXq(i= 1,2) and, consequen ly, p1=p2. This means ha he sequence {π( k, xy)}is con e gen . The second s a emen is he e o e p o ed. Taking in o accoun he i s and second s a emen s, o inish he p oo o he hi d s a emen i is su icien o apply Theo em 4.5. The heo em is comple ely p o ed. Rema k 4.7. 1. Theo em 4.6 emains ue i we eplace he s anda d scala p oduc h·,·i on he space Rnby an a bi a y scala p oduc hu, uiW:= hWu, ui, whe e W= (wij )n i,j=1 (wij ∈R) is a symme ic and posi i e de ined n×n–ma ix. 2. I we eplace condi ion (23) by a s onge condi ion, hen Theo em 4.6 is also ue wi hou he equi emen ha he e exis s a leas one solu ion which is bounded on R+. Namely, i he e exis s a unc ion ζ∈ K such ha h (y, u1)− (y, u2), u1−u2i ≤ −ζ(|u1−u2|2) o all u1, u2∈Rnand y∈Y, whe e ζpossesses some addi ional p ope ies (see, o example, [14]). ON THE STRUCTURE OF THE GLOBAL ATTRACTOR 19 3. I is easy o see ha Theo em 4.6 emains ue also o equa ion (18) in an a bi a y Hilbe space H, i we suppose ha he cocycle ϕ, gene a ed by equa ion (18), is asymp o ically compac (i.e., he co esponding skew-p oduc dynamical sys em is asymp o ically compac ), and we eplace he condi ion abou he exis ence o a leas one solu ion which is bounded on R+, by he exis ence o a ela i ely compac solu ion ϕ0on R+( his means ha ϕ(R+) is a ela i ely compac subse om H). Acknowledgemen s. We would like o hank he anonymous e e ee o he help ul sugges ions which allowed us o imp o e he p esen a ion o his a icle. This pape was w i en while he second au ho was isi ing he Uni e si y o Se illa (Feb ua y–Sep embe 2010) unde he P og ama de Mo ilidad de P o eso es Uni- e si a ios y Ex anje os (Minis e io de Educaci´on, Spain) g an SAB2009-0078. He would like o hank people o his uni e si y o hei e y kind hospi ali y. He also g a e ully acknowledges he inancial suppo o he Minis e io de Educaci´on (Spain). The i s au ho is pa ially suppo ed by g an MTM2008-00088 (Minis- e io de Ciencia e Inno aci´on, Spain) and P oyec o de Excelencia P07-FQM02468 (Jun a de Andaluc´ıa, Spain). REFERENCES [1] N. P. Bha ia and G. P. Szeg¨o, “S abili y Theo y o Dynamical Sys ems”, Lec u e No es in Ma hema ics, Sp inge , Be lin–Heidelbe g–New Yo k, 1970. [2] I. U. B ons eyn, “Ex ensions o Minimal T ans o ma ion G oup”, Noo dho , 1979. [3] B. F. Bylo , R. E. Vinog ad, D. M. G obman and V. V. Nemy skii, “Lyapuno Exponen s Theo y and I s Applica ions o P oblems o S abi y”, Moscow, Nauka, 1966, 576 pp. (in Russian) [4] T. 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