Pullback, forward and chaotic dynamics in 1-D non-autonomous linear-dissipative equations

Pullback, forward and chaotic dynamics in 1-D non-autonomous linear-dissipative equations T. Caraballo1, J.A. Langa1& R. Obaya2 1Departamento de Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080-Sevilla Spain 2Departamento de Matem´atica Aplicada E. Ingenieros Industriales. Universidad de Valladolid. 47011 Valladolid, Spain. Member of IMUVA, Instituto de Matem´aticas Universidad de Valladolid. E-mail: [email protected], [email protected], [email protected] February 2015 Abstract. The global attractor of a skew product semiflow for a non-autonomous differential equation describes the asymptotic behaviour of the model. This attractor is usually characterized as the union, for all the parameters in the base space, of the associated cocycle attractors in the product space. The continuity of the cocycle attractor in the parameter is usually a difficult question. In this paper we develop in detail a 1D non-autonomous linear differential equation and show the richness of non-autonomous dynamics by focusing on the continuity, characterization and chaotic dynamics of the cocycle attractors. In particular, we analyse the sets of continuity and discontinuity for the parameter of the attractors, and relate them with the eventually forward behaviour of the processes. We will also find chaotic behaviour on the attractors in the Li-Yorke and Auslander-Yorke senses. Note that they hold for linear 1D equations, which shows a crucial difference with respect to the presence of chaotic dynamics in autonomous systems. Keywords: global attractor, pullback attractor, forward attractor, 1D non-autonomous linear differential equation, chaotic behavior in Li-Yorke sense, chaotic behavior in Auslander-Yorke sense Mathematics Subject Classification: 37D45, 37A20 1. Introduction We are interested in the asymptotic dynamics of initial value problems of the form (˙x=f(t, x), t > s x(s) = x0∈X, (1) where f:R×D⊂R×X→Xis a map belonging to some metric space C, and Xa Banach space. Assume that, for each f∈ C and x0∈X, the solution of (1) is Continuity and characterization of cocycle attractors 2 defined for all t≥s; that is, for each x0∈X, there is a unique continuous function [s, ∞)3t7→ x(t, s, f, x0)∈Xsatisfying (1). For each t,f(t, ·) is the vector field that drives the solution at time t. Hence, the path described by the solution in Xbetween sand s+τwill depend on both the initial time sand the elapsed time τ. In this paper we assume some kind of recurrence in the temporal variation of the vector fields. In particular, we pay special attention to the almost periodic case. There is a general method to consider the family of non-linearities as a base flow driven by the time shift applied to the non-linearity f(t, ·) of the original equation. We consider f∈Cb(R, X),the set of bounded and uniformly continuous functions from Rinto Xwith the metric ρof the uniform convergence. Denote by P0the set of all translates of f, P0(f) = {f(s+·) : s∈R}, and define the shift operator θt:Cb(R, X)→Cb(R, X) by θtf(·) = f(·+t). For autonomous and periodic time dependence this construction yields a closed base space P0. However, for more general almost-periodic terms it is convenient to consider the closure of P0with respect to ρ: P:= Pρ(f) = closure of P0(f) in Cb(R, X) with respect to ρ, known as the hull of the function fin the space (Cb(R, X); ρ), see [41]. Continuity of θton P0then extends to continuity of θton P. In this paper we consider the 1D linear and dissipative differential equation x0=h(θtp)x+g(x), p ∈P, x ∈R,(2) with ha real almost-periodic function with null mean value and unbounded primitive and P={θth, t ∈R} the hull of h. Note that (P, θ) is a continuous flow in a compact metric space. g:R→R is a smooth function with xg(x)≤0 for all x∈R, limx→∞ g(x) = −∞ and lim|x|→∞ g(x) x=−∞.We denote by C(P) the set of real continuous functions on Pand thus C0(P) will denote the subset of C(P) with null mean value. B(P) will represent the subset of C0(P) with continuous primitive, and U(P) its complementary, i.e., the subset of C0(P) of functions with unbounded primitive (see Section 3). In this framework, two asymptotic behaviours give rise to completely different scenarios. Indeed, asymptotics with respect to time t(uniformly, see Chepyzhov and Vishik [13], or not in s) or with respect to s(when s→ −∞ and tis arbitrary but fixed). These are called, respectively, forward and pullback dynamics and are in general unrelated. Thus, during the last twenty years two main approaches have been developed in order to study attractors for (1): on the one hand, the pullback attractor (Carvalho et Continuity and characterization of cocycle attractors 3 al. [11], Kloeden and Rasmussen [27]), an invariant set for the evolution process which is pullback (but, in general, not forward) attracting; on the other hand, the global attractor for the associated skew-product flow, an invariant compact set attracting forward in time (Sell [42], Kloeden and Rasmussen [27]). The cocycle attractor A(p) (see Definition 3) for (2) is described by an interval [a(p), b(p)],for all p∈P. The aim of this paper is to study in detail the structure and internal dynamics on this family of attractors. Another definition of attractor for nonautonomous dynamical systems is that of the uniform attractor (see Chepyzhov and Vishik [13]) which is then described as the union of all the associated cocycle attractor (see Kloeden and Rasmussen [27], Bortolan et al. [9, 8]) so that, by studying the structure of cocycle attractor, in fact we are also going in detail into the characterization of uniform attractors. A recent related work is Hoang et al. [20], where the authors prove that, given a family of parameterized processes, continuity points (with respect to the parameter) for pullback and uniform attractors is a residual set, so dense in the set of parameters. Our results are different in the sense that our family is given by the driven space of functions p∈P, and not by perturbation of dynamical systems. An important result in Cheban et al. [12] proves that, if the function p→A(p) is upper and lower semicontinuous, then, uniform pullback and uniform forward attraction are equivalent. The results in this paper will confirm that the property of continuity of this set-valued map cannot be weakened. Indeed, in Section 3 we study, for a particular h∈U(P),the set Ps⊂Pof continuity and non-continuity Pf⊂Pof function p→A(p), showing that our attractor is a pinched set (see Definition 1), described as A(p) = 0 for all p∈Psand A(p) = [−b(p), b(p)] with b(p)>0 for all p∈Pf. For a residual set in Ps,we prove (see Proposition 24 and Corollary 26) that there is no forward attraction to A(p),i.e., we lose forward attraction specifically in the continuity points of the cocycle attractor. In some cases this residual set is all Ps.In Section 5 we prove that, generically, this is the situation we find, i.e., if we define Rs(P) = {h∈C0(P) : ν(Ps(h)) = 1} and Rf(P) = {h∈C0(P) : ν(Pf(h)) = 1} with νthe Haar measure on P, we deduce (see Theorem 29) that Rs(P) is a residual set in C0(P).Although topologically more unusual, in Section 5.2 we concentrate in the case when Rf(P)6=∅, so that we can deal with h∈U(P), with ν(Pf(h)) = 1.Theorem 31 proves that we obtain forward attraction in Pf, i.e., we have forward attraction in full measure precisely in the set of non-continuity of the map p→A(p). A recent discussion on forward nonautonomous attractors can be found in Kloeden and Lorenz [28]. In Section 6 we find chaos inside the cocycle attractor. To our knowledge this is the first time in the literature where chaos is studied related to this kind of attractors. Indeed, Theorem 36 shows that, in the previous case with h∈U(P) and ν(Pf(h)) = 1 the sets [−b(p), b(p)] are scrambled (see Definition 33) for a.a. p∈P, leading to Li- Yorke chaotic dynamics in measure (see Blanchard et al. [7]). In addition, we also Continuity and characterization of cocycle attractors 4 obtain explicit examples in the case ν(Ps(h)) = 1 where the cocycle attractor is Li- Yorke chaotic. Finally, in Section 6.2 we can also find sensitive dependence on the set A0=∪p∈P{p} × {0}, so that we also find chaotic dynamics in the Auslander-Yorke sense (see [5]). 2. Basic notions We start with some preliminary concepts and results on topological dynamics and ergodic theory that can be found in Ellis [15], Nemytskii and Stepanov [35], Sell [42] and Shen and Yi [43]. Let (P, dP) be a compact metric space and θ={θt}t∈Ra real continuous flow on P. Given p∈P, the set {θtp}t∈Ris called the orbit of p. We say that a subset P1⊂Pis θ-invariant if θt(P1) = P1for all t∈R.A subset P1is minimal if it is compact invariant and it does not contain properly any other compact invariant set. We say that the continuous flow (P, θ) is recurrent or minimal if Pis minimal. A normalized regular measure νdefined on the Borel sets of Pis invariant if ν(θt(P1)) = ν(P1) for every Borel subset P1⊂Pand every t∈R.It is ergodic if, in addition, ν(P1) = 1 or ν(P1) = 0 for every invariant subset P1.The set of normalized invariant measures is not void. We say that (P, dP) is uniquely ergodic if it has a unique normalized invariant measure which is necessarily ergodic. We say that the flow (P, dP) is almost-periodic if the family {θt}t∈Rof section maps is equicontinuous, i.e., for every ε > 0 there exists δ > 0 such that if p1, p2∈Pand dP(p1, p2)< δ then dP(θtp1, θtp2)< ε for every t∈R. A subset L⊂Ris said to be relatively dense if there exists a number l > 0 such that every interval [r, r +l] contains at least a point of L. We say that f∈Cb(R,R) is almost periodic if for every ε > 0 there exists a relatively dense subset Lε(f) such that supt∈R|f(t+r)−f(t)| ≤ εfor every r∈Lε(f).If f∈Cb(R,R) is almost-periodic then the hull P=P(f) of fis a compact metric space and if {θt}t∈Rdenotes the shift operator, then the flow (P, θ) is almost-periodic, minimal and uniquely ergodic. In fact Pis an abelian topological group and the Haar measure is its only invariant measure. We introduce two types of almost-periodic functions that will play a relevant role in what follows. Let α= (α1, . . . , αm)∈Rmbe a vector with rational independent components. The Kronecker flow of vector αis defined on the m−dimensional torus Tmby the map θα:R×Tm→Tm, (t, x1, . . . , xm)→(x1+tα1, . . . , xm+tαm),and it is almost periodic and minimal (see [15, 43]). We say that a function f∈C(R,R) is quasi-periodic if there exists a Kronecker flow (Tm, θα) and a function h∈C(Tm) with f(t) = h(α1t, . . . , αmt) for every t∈R.Under this condition the hull of fis isomorphic to a k-dimensional torus of (Tm, θα). We say that a function h∈C(R,R) is limit-periodic if is the uniform limit of a sequence of continuous and periodic functions. In this case the hull of hhas frequently a more complicated structure: in simple cases it provides a solenoid. Many relevant examples in the literature considered in this paper have been developed by quasi-periodic Continuity and characterization of cocycle attractors 5 or limit-periodic functions. We can try to analyse non-autonomous differential equations (1) as the combination of a base flow {θt}t∈Ron Pand, for each p∈P, the semiflow R+×X3(t, x0)7→ ϕ(t, p)x0∈Xwhere, for each x0∈X,R+3t7→ ϕ(t, p)x0∈Xis the solution of the initial value problem (˙x=p(t, x), t > 0, x(0) = x0∈X. (3) Then, the family of mappings (t, p)∈R+×p7→ ϕ(t, p)∈ C(X), satisfies •ϕ(0, p) = IdXfor all p∈P, •x7→ ϕ(t, p)x∈Xis continuous, and •for all t≥s,s∈R,and p∈P, ϕ(t+s, p) = ϕ(t, θsp)ϕ(s, p), the ‘cocycle property’. One interprets ϕ(t, p)xas the solution at time tthat has started in the state xat time zero subjected to the non-autonomous driving term p∈P. The pair (ϕ, θ)(X,P )will be called a non-autonomous dynamical system on (X, P) (see Kloeden and Rasmussen [27]). Now, given a non-autonomous dynamical system (ϕ, θ)(X,P ), one can also define an associated autonomous dynamical system (see [41, 42]) Π(·) on X=P×X(with the metric dX((x, p),(¯x, ¯p)) = d(x, ¯x)+dP(p, ¯p)) by setting Π(t)(p, x) = (θtp, ϕ(t, p)x)), t ≥0. The semigroup property of θtand the cocycle property of ϕensure that Π(·) satisfies the semigroup property. Thus, given a non-autonomous differential equation such as (1), we need to deal with four different dynamical systems: (a) The driving semigroup {θt:t≥0}on Passociated to the dynamics of the timedependent nonlinearities appearing in the equation. (b) the skew-product semiflow {Π(t) : t≥0}defined on the product space P×X, (c) the associated non-autonomous dynamical system (ϕ, θ)(X,P )with ϕ(t, θsf)x0= x(t+s, f, x0), (d) and the evolution process S(t, s)x0=u(t−s, θsf)x0. Observe that these dynamical systems can possess an associated attractor: (i) A global attractor Afor the skew-product semiflow Π(t), Continuity and characterization of cocycle attractors 6 (ii) a cocycle attractor {A(p)}p∈pfor the cocycle semiflow ϕ, (see Kloeden and Rasmussen [27]) (iii) a pullback attractor {A(t)}t∈Rfor the evolution process S(t, s) (see Carvalho et al. [11]). We next introduce and compare some concepts of the topological and random theory of dynamical systems. In this paper we always assume that the base flow (P, θ, R) is mininal. We first consider some topological notions. The concepts of minimal and invariant measure admit natural extensions for semiflows. Definition 1. (i) A minimal set K⊂P×Xis called an automorphic extension of the base Pif, for some p∈P,K∩Π−1 P(p)is singleton, with ΠPthe projection on the first component of P×X. In these conditions we say that the minimal set Kis almost-automorphic when the flow on the base Pis almost-periodic. (ii) A compact invariant set K⊂P×Xis called a pinched set if there exists a residual set P0(Psuch that K∩Π−1 P(p)is a singleton for all p∈P0and K∩Π−1 P(p) is not a singleton for all p /∈P0. Note that an invariant compact set K⊂P×Xis almost automorphic if it is pinched and minimal. Given a NDS (ϕ, θ)(X,P ), suppose that the associated skew–product semiflow semigroup {Π(t) : t⩾0}possesses a global attractor Aon P×X. We know that {Π(t) : t⩾0}has a global attractor if and only if there exists a compact set K⊂P×X such that lim t→∞ dist(Π(t)B,K)=0,(4) for any bounded subset Bof P×X, where dist denotes the Hausdorff semidistance between sets defined as dist(A, B) = sup a∈A inf b∈bd(a, b). Definition 2. (i) A non-autonomous set is a family {D(p)}p∈Pof subsets of Xindexed in p. We say that {D(p)}p∈Pis an open (closed, compact) non-autonomous set if each fiber D(p)is an open (closed, compact) subset of X. (ii) A non-autonomous set {D(p)}p∈Pis invariant under the NDS (ϕ, θ)(X,P )if ϕ(t, p)D(p) = D(θtp), for all t⩾0and each p∈P. It is immediate that a non-autonomous set {D(p)}p∈Pis invariant for (ϕ, θ)(X,P )if and only if the corresponding subset Dof P×X, given by D=[ p∈P {p} × D(p), is invariant for the semigroup {Π(t) : t⩾0}. Continuity and characterization of cocycle attractors 7 Given a subset Eof P×Xwe denote by E(p) = {x∈X: (x, p)∈E}the p−section of E; hence E=[ p∈p {p} × E(p) (5) Given a non-autonomous set {E(p)}p∈Pwe denote by Ethe set defined by (5). Note that [ p∈p E(p)=ΠXE, where we denote by ΠXthe projection on the second component in P×X. We can now relate the concept of cocycle attractors for (ϕ, θ)(X,P )with the global attractor for the associated skew–product semiflow {Π(t) : t⩾0}. Definition 3. Suppose Pis compact and invariant and that {θt:t∈R}is a group over Pand θ−1 t=θ−t, for all t > 0. A compact non-autonomous set {A(p)}p∈Pis called a cocycle attractor of (ϕ, θ)(X,P )if (i) {A(p)}p∈Pis invariant under the NDS (ϕ, θ)(X,P ); i.e., ϕ(t, p)A(p) = A(θtp),for all t⩾0. (ii) {A(p)}p∈Ppullback attracts all bounded subsets B⊂X, i.e., for all p∈P, lim t→+∞dist(ϕ(t, θ−tp)B, A(p)) = 0. The following result can be found, for instance, in Propositions 3.30 and 3.31 in Kloeden and Rasmussen [27], or Theorem 3.4 in Caraballo et al. [10]. Theorem 4. Let (ϕ, θ)(X,P )be a non-autonomous dynamical system, where Pis compact, and let {Π(t) : t⩾0}be the associated skew–product semiflow on P×X with a global attractor A. Then {A(p)}p∈Pwith A(p) = {x∈X: (x, p)∈A}is the cocycle attractor of (ϕ, θ)(X,P ). The following result offers a converse (see Proposition 3.31 in [27], or Lemma 16.5 in [11]). Theorem 5. Suppose that {A(p)}p∈Pis the cocycle attractor of (ϕ, θ)(X,P ), and {Π(t) : t⩾0}is the associated skew–product semiflow. Assume that {A(p)}p∈Pis uniformly attracting, i.e., there exists K⊂Xcompact such that, for all B⊂Xbounded, lim t→+∞sup p∈P dist(ϕ(t, θ−tp)B, K) = 0, and that Sp∈PA(p)is precompact in X. Then the set Aassociated with {A(p)}p∈P, given by A=[ p∈P {p} × A(p), is the global attractor of the semigroup {Π(t) : t⩾0}. Continuity and characterization of cocycle attractors 8 3. Non-uniform cocycle attractors Let (P, θ, R) be a minimal flow on a compact metric space P. For a given Banach space Xwe consider a skew-product semiflow {Π(t)}t∈R+on P×X. Suppose Π(t) admits a global attractor Adescribed by A=[ p∈P {p} × A(p). In Cheban et al. [12] it is proved that the continuity of the set-function p→A(p) implies the uniform pullback, and therefore uniform forward, attraction to the cocycle attractor A(p) given by Theorem 4. The aim of the following sections is to develop some non-trivial models in which the above function is not continuous in the whole P, and, by a careful study of its sets of continuity, to give a detailed description on the dynamics and the structure of the attractors. 3.1. Attractors for order preserving non-autonomous systems In what follows we suposse Xis a partially ordered Banach space, i.e. there exists a closed convex positive cone X+⊂X, which is also a vectorial subspace of X, such that X+∩(−X+) = {0}. This set X+defines a partial order relation on Xin the way x≤yif y−x∈X+; we write x<yif x≤yand x6=y. If in addition int(X+)6=∅we say that Xis strongly ordered. Definition 6. Let (ϕ, θ)(X,P )be a non-autonomous dynamical system. We say that ϕ is order-preserving for the order relation ‘≤’ in Xif u0≤v0implies that ϕ(t, p)u0≤ ϕ(t, p)v0,for all p∈Pand t≥0. In this section we assume that the non-autonomous dynamical system (ϕ, θ) generated by equation (1) is order preserving for the order in X. We introduce the concepts of sub–, super– and equilibrium given by Arnold and Chueshov [3] in the stochastic setting (see also Chueshov [14]) and by Novo et al. [34] in the topological framework. Definition 7. A Borel map a:P→Xsuch that ϕ(t, p)a(p)is defined for any t≥0is said to be a) an equilibrium if a(θtp) = ϕ(t, p)a(p),for any p∈Pand t≥0, b) a super-equilibrium if a(θtp)≥ϕ(t, p)a(p),for any p∈Pand t≥0, c) a sub-equilibrium if a(θtp)≤ϕ(t, p)a(p),for any p∈Pand t≥0. Definition 8. A super-equilibrium (resp. sub-equilibrium) a:P→Xis semicontinuous if the following holds i) Γa= closureX{a(p) : p∈P}is a compact subset in X; Continuity and characterization of cocycle attractors 9 ii) Ca={(p, x) : x≤a(p)}(resp. Ca={(p, x) : x≥a(p)}) is a closed subset of P×X. An equilibrium is semi–continuous if it holds i) and ii) above. We name a semiequilibrium to a sub-equilibrium or a super-equilibrium. The following result, that will be relevant in the topological version of the semiequilibria, was proved in Proposition 3.4 of Novo et al. [34], following classical arguments from Aubin and Frankowska [4]. Proposition 9. Assume that a:P→Xis a semi-continuous semi-equilibrium. Then it possesses a residual invariant set Pcof continuity points. We assume that ϕadmits a cocycle attractor. The following result provides sufficient conditions for the existence of upper and lower asymptotically stable semiequilibria, giving some useful information on the structure of this invariant set. The proof was given by Arnold and Chueshov [3] in the random context and generalized to the topological formulation in Novo et al. [34]. Theorem 10. Let ϕbe an order-preserving process and A(p)be its associated cocycle attractor. Suppose there exist Borel maps α, β :P→Xsuch that the cocycle attractor is in the “interval” [α(p), β(p)], i.e. A(p)⊂Iβ α(p) = [α(p), β(p)] = {x∈X:α(p)≤x≤β(p)}. Then, there exist two equilibria a, b :P→Xwith a(p), b(p)∈A(p)such that i) α(p)≤a(p)≤b(p)≤β(p),and A(p)⊂Ib a(p),for all p∈P. ii) ais minimal (bis maximal) in the sense that it does not exist any complete trajectory in the interval Ia α(Iβ b). iii) a(p)is pullback asymptotically stable from below, that is, for all v(·)with α(p)≤ v(p)≤a(p),for all p∈P, we have that lim t→+∞d(ϕ(t, θ−tp)v(θ−tp), a(p)) = 0. b(p)is pullback asymptotically stable from above, that is, for all v(·)with β(p)≥ v(p)≥b(p),for all p∈P, we have that lim t→+∞d(ϕ(t, θ−tp)v(θ−tp), b(p)) = 0. iv) If A=Sp∈P{p} × A(p)is compact and the maps α, β are continuous, then the functions p→a(p), p →b(p)are semi-continuous and admits a residual set Pc⊂P of points of continuity. v) Assume condition in iv), and take p0∈Pc.Then the sets Ka={(θtp0, a(θtp0), t ∈R} and Kb={(θtp0, b(θtp0), t ∈R} Continuity and characterization of cocycle attractors 16 Definition 20. A function h∈C0(P)is Kozlov if every p∈Pis recurrent for h. We consider h∈C0(p) and the above framework for (10). Then there exists Pf, invariant and of first category, and its complementary, the residual set Ps,such that the cocycle attractor A(p) = [−b(p), b(p)] with b(p)>0 if p∈Pfand b(p) = 0 if p∈Ps. Let Prbe the recurrent points and Pathe asymptotic points. Recall that we denote by Pothe oscillatory points in P. We firstly have the following result Proposition 21. Let p0∈P. i) sup t≤0Zt 0 h(θsp0)ds =∞if and only if b(p0) = 0,i.e., p0∈Ps, and ii) sup t≤0Zt 0 h(θsp0)ds < ∞if and only if b(p0)>0,i.e., p0∈Pf. iii) If lim sup t→−∞ Zt 0 h(θsp0)ds < sup t∈RZt 0 h(θsp0)ds then p0∈Psand there exists t∈Rwith b(θtp0)> r0. Proof. i) Let yp0(t) = exp(Rt 0h(θsp0)ds).Note that yp0(t) = y(t, p0; 1) is the solution of (7) with yp0(0) = 1. Then there exists tn→ ∞ with lim n→∞ Z−tn 0 h(θsp0)ds → ∞.Suppose rbig enough. For T > 0, bT(p0)≤y(T, p0(−T), r),and b(p0) = limT→∞ bT(p0).We have y(tn, p0(−tn); r) = r yp0(−tn)y(tn, p0(−tn); yp0(−tn)) = r yp0(−tn) converges to zero as n→ ∞, which implies the equivalence with b(p0)=0,i.e. p0∈Ps. For ii), let ρ > 0 with sup t≤0 ρ eRt 0h(θsp0)ds ≤r0. Then, if x(t;p0;ρ) is the solution of (10) with x(0) = ρit holds x(t, p0;ρ) = ρ eRt 0h(θsp0)ds for all t≤0. On the other hand, since {x(t, p0;ρ) : t∈R}is bounded, it is on the cocycle attractor, i.e. [0, ρ]⊂[0, b(p0)] and then b(p0)>0. For iii), it is clear that supt≤0Zt 0 h(θsp0)ds < ∞and hence p0∈Pf.There are t1< t2with supt≤t1Zt 0 h(θsp0)ds < Zt2 0 h(θsp0)ds =ρfor every t≤t1.Let y(t) = y(t, p0;r0 ρ) = r0 ρexp(Rt 0h(θsp0)ds) be the solution of (7). Let t3∈(−∞, t2] Continuity and characterization of cocycle attractors 17 be the first point with y(t3) = r0.There exists γ > 1 with y(t, θt1p0;γy(t1)) ≤r0for every t≤t1.Then the solution of the nonlinear equation (10) satisfies x(t, θt1p0;γy(t1)) = y(t, θt1p0;γy(t1)) for every t≤t1and x(t3−t1, θt1p0;γy(t1)) > x(t3−t1, θt1p0, y(t1)) = y(t3−t1, θt1p0, y(t1)) = y(t3, p0;r0 ρ) = r0. Hence b(θt3p0)≥x(t3−t1, θt1p0;γy(t1)) > r0. Note that in this case the function t→b(θtp0) is not solution of the linear equation (7) as its affected by the dissipation term in (10). From this last result we deduce Corollary 22. Po⊂Ps. The following result characterizes the forward attraction in the cocycle attractor. Proposition 23. Let p0∈Pand x0∈R.Then it holds lim t→∞ x(t, p0;x0) = 0 if and only if p0∈Pa. Proof. Suppose there exists t0such that if t≥t0then x(t, p0;x0)≤r0.Then x(t, p0;x0) = x(t−t0, θt0p0;x(t0, p0;x0)) = x(t0, p0;x0)eRt−t0 0h(p0(t0+s))ds, which implies lim t→∞ Zt 0 h(θsp0)ds =−∞, so that p0∈Pa. On the other hand, 0< x(t, p0;x0)≤x0eRt 0h(θsp0)ds which tends to zero for every x0∈Rif p0∈Pa. Proposition 24. Let p0∈P. i) If p0∈Pa,the process ϕ(t, p0)has a forward attractor defined by {0}. ii) If p0∈Ps∩P+ r,the process ϕ(t, p0)has no forward attractor. Proof. i) and ii) are clear from Proposition 23. Indeed, if we have forward attraction to zero we have that p0∈Pa. Continuity and characterization of cocycle attractors 18 Remark 25. Note that if p0∈Pa∩Pfwe have proved that b(p0)>0and limt→∞ b(θtp0)=0.Thus, a proper definition of a forward attractor {A(p)}p∈Pfor the cocycle should consider minimality of the family A(p), in the sense that there is no proper invariant forward attracting family included in A(p).Thus, for p0∈Pa∩Pfthe forward attractor should be defined as the constant family A(p)=0. The following results are immediate consequences of Proposition 24. Corollary 26. If his Kozlov then there is no forward attractor in Ps. Corollary 27. If h∈U(P)and ν(Ps) = 1 then there exists a residual set of full measure, P∗ ssuch that, if p0∈P∗ sthe process ϕ(t, p0)has no forward attractor. 5. The sets Rs(P), Rf(P).Genericity of ν(Ps)=1. In the rest of the paper we will represent by Ps(h), Pf(h) the invariant subsets defined in the previous section to emphasize their dependence with respect to h∈C0(P).We define the sets Rs(P) = {h∈C0(P) : ν(Ps(h)) = 1} and Rf(P) = {h∈C0(P) : ν(Pf(h)) = 1}. It is clear that B(P)⊂Rs(P) and that Rs(P)∪Rf(P) = C0(P).In this section we analyse the topological size of these sets in C0(P). 5.1. Rs(P)is residual in C0(P). We consider the time reversed flow ˆσon R×Pdefined as ˆσ(t, p) = θ−tp. If y(t) is a solution of (7) through p0∈Pthen ˆy(t) = y(−t) satisfies ˆy0(t) = −h(θ−tp0)ˆy(t).For simplicity we denote by ˆ Pthe base space with time reversed flow, i.e. ˆ P= (P, ˆσ, R).Note that the reverse of the flow ˆσis again σ. Proposition 28. It holds i) For any h∈C0(P),either h∈Rs(P)or −h∈Rs(P). ii) For the time-reversed flow, Rs(ˆ P) = −Rs(P), Rf(ˆ P) = −Rf(P). Proof. Let h∈U(P) and fix p0∈P. If supt≤0Zt 0 h(θsp0)ds ≤ ∞, then it follows from Proposition 12 that inft≤0Zt 0 h(θsp0)ds =−∞,so that supt≤0Zt 0 −h(θsp0)ds =∞and p0∈Ps(−h).As a consequence of the ergodicity of νwe conclude that at least either Continuity and characterization of cocycle attractors 19 ν(Ps(h)) = 1 or ν(Ps(−h)) = 1. For ii), suppose that we are in the case with ν(Ps(h)) = 1 and take p0∈Ps(h),so that sup t≤0Zt 0 h(θsp0)ds =∞. Then there exists a sequence t1 n→ −∞ with Zt1 n 0 h(θsp0)ds =∞. As ν(Pr) = 1 we can suppose that all the points of the sets {θtp0:t∈R}are recurrent points. For each n∈Nthere exists t2 n>0 such that the sequence Zt2 n t1 n h(θsp0)ds =Zt2 n−t1 n 0 h(θ(t1 n+s))p0ds →0, this last property by the recurrence of path. But note that Zt2 n t1 n h(θsp0)ds =−Zt1 n 0 h(θsp0)ds +Zt2 n 0 h(θsp0)ds so that, as Zt1 n 0 h(θ(t1 n+s))p0ds → −∞, then supt≥0Rt 0h(θsp0)ds =∞. We consider the time reversed flow ˆσon R×P. Since, for t > 0 Zt 0 h(θsp0)ds =Z−t 0 −h(θ−sp0)ds, it holds sup t≤0Zt 0 −h(θ−sp0)ds = sup t≥0Zt 0 h(θsp0)ds =∞ hence p0∈ˆ Ps(−h) and then we have that ν(ˆ Ps(−h)) = 1 and −h∈Rs(ˆ P).As a consequence Rs(P)⊂ −Rs(ˆ P).A symmetric argument proves that Rs(ˆ P)⊂Rs(P) and thus equality. Now it is also straightforward that Rf(ˆ P) = −Rf(P) Observe that this last result shows how big Rs(P) is on C0(P), since it shows that C0(P) = Rs(P)S(−Rs(P)). The references Johnson [25] and Novo and Obaya [33] provide criteria for existence and precise examples of functions h∈U(P) and L:P→Rmeasurable with L(θtp)−L(p) = Rt 0h(θsp)ds for almost every p∈Pand t∈R.We say that L is a measurable primitive along the flow on h. The examples are based in previous construction given in Furstenberg [16]. The condition h∈Rf(P) requires in addition that eL∈L∞(P).The example 3.2.1 in Johnson [23] uses methods, already suggested in Anosov [2], to construct quasi period flows in the 2D torus T2and a function h∈C0(T2) with L(p) = sup t∈RZt 0 h(θsp)ds ≤L0<∞a.e. Continuity and characterization of cocycle attractors 20 In this case h∈Rf(P) and, moreover, for a.a. p∈P L(θtp)−L(p) = Zt 0 h(θsp)ds for all t∈R. This method was improved in the Appendix of Ortega and Tarallo [36], which in particular implies that this kind of function hexists for every quasi-periodic flow. Theorem 29. i) Rs(P)is a residual set in C0(P). ii) The set Ro(P) = {h∈C0(P) : ν(Po(h)) = 1)}is also residual in C0(P). Proof. For h∈C0(P), k∈N, k ≥1 we define Nk(h) = {p∈P: lim sup T→∞ 1 2Tl({t∈[−T, T] : Zt 0 h(θsp)ds ≤k}) = 0}, with lthe Lebesgue measure in R.It is clear that Nk+1(h)⊂Nk(h) for all k∈N.In Johnson [24] it is proved that the set C∗ 0={h∈C0(P) : ν(Nk(h) = 1 for all k∈N} is residual in C0(P).If h∈Rf(P) then ν(Pf(h)) = 1,i.e. L(p) = sup t∈RZt 0 h(θsp)ds < ∞a.e. and hence for a.e. p∈Pand all t∈R L(θtp)−L(p) = Zt 0 h(θsp)ds. For kbig enough it is easy to prove that lim sup T→∞ 1 2Tl({t∈[−T, T] : Zt 0 h(θsp)ds ≤k})>0 for a subset of Pwith positive measure. Thus, for klarge enough, ν(Nk(h)) = 0,so that h∈C0(P)\C∗ 0,i.e., C∗ 0⊂Rs(P) and this set is residual. For ii), let R∗be the set of functions h∈U(P) satisfying sup t≤0Zt 0 h(θsp0)ds = sup t≥0Zt 0 h(θsp0)ds =∞for a.a. p∈P, inf t≤0Zt 0 h(θsp0)ds = inf t≥0Zt 0 h(θsp0)ds =−∞ for a.a. p∈P. Then R∗=Rs(P)T(−Rs(P)) is residual, which implies that Ro(P) = {h∈U(P) : ν(P0(h)) = 1}is a residual set in C0(P). Continuity and characterization of cocycle attractors 21 5.2. The case Rf(h)) 6=∅. In this section suppose there exists h∈U(P) with ν(Pf(h)) = 1.Then it holds Proposition 30. Rf(P)is a dense first category set in C0(P). Proof. From the last result, it is clear that Rf(P) is of first category. Fix h∗∈Rf(P). Then {h+ρh∗:h∈B(P), ρ > 0} ⊂ Rf(P). Fix h∈C0(P) and ε > 0. For ||·|| the supreme norm on C0(P).There exist h0∈B(P), ρ0>0 with ||h−h0|| < ε/2,and ρ0||h∗|| < ε/2.Then h0+ρ0h∗∈Rf(P) and ||h−h0−ρ0h∗|| < ε. In Section 3.4 we have shown the existence of a cocycle attractor defined by a pinched set, which is continuous in parameter pwhenever p∈Ps, and which is not forward attracting in the residual set Ps. The following result gives a forward attraction to the cocycle attractor in a set of non-continuity and of full measure. Note that, from the result in Cheban et al [12] one could tend to think that the forward attraction in a cocycle attractor is related to the continuity in the parameter for the cocycle attractor. The following result shows that the uniformity condition for the continuity in [12] is necessary. Theorem 31. Let h∈C0(P)\B(P),with ν(Ps(h)) = 0.Then there exists an invariant set Pfa⊂Pf(h)with ν(Pfa) = 1 such that if p∈Pfathen A(p)is the forward attractor of the process ϕ(t, θsp)x0=x(t−s, θsp;x0)associated to (10). Proof. For p∈Pf(h) we have that A(p) = [a(p), b(p)] with a(p)<0< b(p).Moreover, for rbig enough, b(p) = lim T→∞ bT(p) = lim T→∞ x(T, θ−Tp;r); a(p) = lim T→∞ aT(p) = lim T→∞ x(T, θ−Tp;−r). By Egorov’s theorem (Rudin [38]) there exists a compact set Pf0⊂Pf(h) with ν(Pf0)>0 (as close to one as desired) such that b(p) = lim T→∞ bT(p),uniformly in Pf0. Thus, bis continuous in the compact set Pf0and then there exists δ > 0 with b|Pf0≥δ > 0.Let λ≥1.We now prove that λg(b(p)) ≥g(λb(p)).Indeed, if g(b(p)) = 0 is clear. If g(b(p)) = −(b(p)−r0)2then g(λb(p)) = −(λb(p)−r0)2≤ −λ2(b(p)−r0)2≤ −λ(b(p)−r0)2=λg(b(p)).Thus, (λb(p))0=λb(p)h(p) + λg(b(p)) ≥h(p)λb(p) + g(λb(p)), which means that λb(p) is a super-equilibrium for (10). Thus, if λ > 1 and p∈P b(θtp)≤x(t, p;λb(p)) ≤λb(θtp),for all t≥0. Continuity and characterization of cocycle attractors 22 By Birkhoff’s Ergodic Theorem (Nemytskii and Stepanov [35]) there exists an invariant set Pfawith ν(Pfa) = 1 such that for all p∈Pfathere exists a sequence {t∗ n}n∈Nwith t∗ n→ ∞ and θtp∗ n∈Pf0.We will prove that for p∈Pfaand r > r0big enough we have that lim t→∞(x(t, p;r)−b(θtp)) = 0. Let ε > 0 and λ > 1 with b(p)(λ−1) ≤εfor all p∈P. For p∈Pfa,there exists a t∗ n with θtp∗ n∈Pf0satisfying bt∗ n(θtp∗ n)≤λb(θtp∗ n) by the uniform convergence in Pf0,hence 0≤bt∗ n(θtp∗ n)−b(θtp∗ n)≤(λ−1)b(θtp∗ n)≤ε. Then, if t≥t∗ n x(t, p;r) = x(t−t∗ n, θtp∗ n;x(t∗ n, p;r)) = x(t−t∗ n, θtp∗ n;bt∗ n(θtp∗ n)) ≤x(t−t∗ n, θtp∗ n;λb(θtp∗ n)) ≤λb(t−t∗ n, θtp∗ n;b(θtp∗ n)) = λb(θtp). Then, for all t≥t∗ n, 0≤x(t, p;r)−b(θtp)≤(λ−1)b(θtp)≤ε, which implies the forward convergence in Pfa. Remark 32. Note that in this case we have obtained that the cocycle attractor A(p)6= {0}with full measure (as ν(Pfa) = 1) in a subset of no continuity points for the cocycle attractor, in which we also find forward attraction. We see that is a natural fact not to obtain forward convergence where the cocycle attractor is continuous (see also [28] for a related example on this fact). 6. Chaotic dynamics on the attractor In this last section we study in detail the dynamical complexity of cocycle attractors. We show the presence of different types of chaotic behaviour in our cocycle attractor. In particular, we prove that the attractor possesses chaotic dynamics in the Li-Yorke sense, and that there exists sensitive dependence on initial conditions. 6.1. Chaotic cocycle attractors in the Li-Yorke sense In this final section we will study chaotic dynamics in the Li-Yorke sense on our cocycle attractors, introduced in [31]. Important consequences of this chaotic behaviour can be found in Blanchard et al. [7]. Definition 33. Given (K, σ, d)a continuous flow in a compact metric space, a pair {x, y} ⊂ Kis called a Li-Yorke pair if it holds lim sup t→∞ d(σ(t;x), σ(t;y)>0,lim inf t→∞ d(σ(t;x), σ(t;y)) = 0. Continuity and characterization of cocycle attractors 23 A set S⊂Kis said to be scrambled if every {x, y} ⊂ Sis a Li-Yorke pair. Finally, we say that the flow (K, σ, d)is chaotic in the Li-Yorke sense if there exists an uncountable scrambled S⊂K. We will now consider our cocycle attractor A(p)=[a(p), b(p)] associated to (10) and consider K=A=[ p∈P {p} × [a(p), b(p)]. Since our flow on the base (P, σ, R) is almost-periodic it is obvious that if (p1, x1)∈P×R, (p2, x2)∈P×Rare a Li-Yorke pair then p1=p2.Thus, if S0⊂P×Ris a scrambled set there exists p0∈Psuch that S0⊂ {p0}×A(p0).This motivates the following definition: Definition 34. We say that Ais fiber-chaotic in measure in the Li-Yorke sense if there exists an invariant set Pch ⊂Pwith ν(Pch)=1such that {p} × A(p)is scrambled for all p∈Pch. Note that Pch ⊂Pfand it is a set of first category. Thus, our set is different from the residually Li-Yorke chaotic sets analysed in Bjerklov and Johnson [6] and Huand and Yi [21]. The arguments of these papers also shows that our fiber-chaotic compact set has zero topological entropy. 6.1.1. Chaotic dynamics with full measure We consider the framework of the previous section, that is, we have ν(Pf) = 1 being b(p)>0 for all p∈Pf. We first need the following important result which guarantees that, with full measure, the cocycle attractor is described from complete bounded trajectories of the linear system (7). Theorem 35. There exists Pl⊂Pfinvariant and with ν(Pl) = 1 such that 0< b(p)≤r0 for all p∈Pl. Proof. Let us define D0={p∈P: there exists t∈Rwith b(θtp)> r0}.It is clear that D0is measurable and invariant. We argue by contradiction and assume that ν(D0) = 1. Take m∈Nand Dm={p∈P: there exists t∈Rwith b(θtp)> r0+1 m}.Note that D0=S∞ m=1 Dm.Then there exists m0∈Nwith ν(Dm0)>0.Define D+ m={p∈P: there exists t > 0 with b(θtp)> r0+1 m}. Let E0⊂Dm0compact with ν(E0)>0,and consider the restriction of bto E0,b|E0 continuous. Birkhoff ergodic theorem assures the existence of a compact set E1⊂E0 with 0 < ν(E1)<1 such that for all p∈E1there exist sequences s1 n→ ∞, s2 n→ −∞ (depending on p) such that θsp1 n, θsp2 n∈E0for all n∈N.Note that for all p∈E0there exists t(p) with b(θtp(p)) > r0+1 m0.Since b(θtp) = x(t, p;b(p)) for every p∈Pand b is continuous on E0, then b(θtp1(p)) > r0+1 m0for all p1∈B(p, δ(p)) ∩E0for suitable δ(p)>0. Continuity and characterization of cocycle attractors 24 Finally, E0⊂ ∪p∈E0B(p, δ(p)) admits, by compactness, a recovering by a finite number of sets, so that there exists T0>0 such that, for all p∈E0we find t(p) with |t(p)| ≤ T0satisfying b(θtp)> r0+1 m0. From here it is easy to prove that E1⊂D+ m.If p∈E1there are s1=s(p)> T0and t1=t(θsp1) with |t1|< T0, θsp1∈E0and such that b(θs1+t1p)> r0+1 m.This implies that p∈D+ m. If we now denote as usual by y(t, p;x0) the solution of the linear equation (7) through pwith y(0) = x0,we will prove that, for each fixed p1∈E1, then limn→∞ y(s1 n, p1;b(p1)) = ∞,where s1 n=s1 n(p1).Denote sn=s1 n.We can suppose that sn+1 −sn≥T0+ 1 for every n∈N.We argue by contradiction and suppose also that θsnp1∈E0tends to p∗∈E0and limn→∞ y(sn, p1;b(p1) = γ0b(p∗)<∞. Note that if λ≥0 (λb(θtp))0=h(θtp)λb(θtp) + λg(b(θtp)) ≤h(θtp)λb(θtp),for all t∈R, p ∈P i.e., γ0b(p) is a sub-equilibrium for (10) and, for all t≥0, p∈P y(t, p;γ0b(p)) ≥γ0b(θtp). Moreover, for p∗there exists t(p∗)>0 with b(θt(p∗)p∗)> r0+1 mand hence γ0g(b(θt(p∗)p∗)) >0,then d dt(γ0b(θtp∗))|t=t(p∗)< h(θtp∗)γ0b(θtp∗)|t=t(p∗), implying that the sub-equilibrium is strong in the sense we next explain. There exist γ2> γ1> γ0and t0> t∗(p∗)>0 with y(t, p∗;γ0b(p∗)) ≥γ2b(θtp∗) for every t≥t0.Moreover b(θt0p∗) = lim n→∞ b(θsn+t0p1) hence there exists n0∈Nsuch that y(sn0+t0, p1;b(p1)) ≥y(t0, θsn0p1;y(sn0, p1;b(p1)) ≥y(t0, θsn0p1;γ0b(θsn0p1)) > γ1b(θsn0+t0p1) and thus y(sn, p1;b(p1)) ≥γ1b(p∗) if sn≥sn0+t0.Finally lim n→∞ y(sn, p1;b(p1)) ≥γ1b(p∗), which contradicts the definition of γ0. As a consequence lim n→∞ Zsn 0 h(θsb(p))ds =∞. But, application of Theorem 18 shows that supt≤0Rt 0h(θsb(p))ds =∞for a.e. p∈E1, which is impossible as ν(Pf(h)) = 1.Thus, ν(D0) = 0 and the result is proved. Continuity and characterization of cocycle attractors 25 Let E⊂Plbe a compact set such that ν(E)>0 and the restriction b|Econtinuous. Let E∞={p∈Pl: there exists tn→ ∞ with θtpn∈E}. We know that ν(E∞) = 1.We will prove that Theorem 36. For all p∈E∞,the sets {p} × [−b(p), b(p)] is scrambled. Proof. Note that it is enough to prove it for {p} × [0, b(p)].Take p∈E∞and p0∈Ps. Then there exist sequences t1 n, t2 nwith θtp1 n∈Efor all n∈Nand θtp2 n→p0∈Ps. Then, if x1, x2∈(0, b(p)] there exist γ1, γ2∈(0,1), γ16=γ2, such that x1=γ1b(p) and x2=γ2b(p).Then |x(t1 n, p;γ1b(p)) −x(t1 n, p;γ2b(p))|=|γ1−γ2|b(p)eZt1 n 0 h(θsp)ds =|γ1−γ2|b(θtp1 n) which is between δ|γ1−γ2|and γ|γ1−γ2|for some δ, γ > 0 by the continuity of bon the compact set E. In the same way |x(t2 n, p;γ1b(p)) −x(t2 n, p;γ2b(p))| ≤ b(θtp2 n)→0. Note that the result is also true if γ1= 0. The compact Sp∈Pl{p} × [−b(p), b(p)] ⊂P×Ris also invariant for the linear flow defined by (7), so that the above result shows the restriction of the flow is Li-Yorke chaotic. 6.1.2. Chaotic dynamics in a fiber In this final section we prove the existence of chaotic dynamics in the Li-Yorke sense in some cases where ν(Pf)=0. Theorem 37. Let h∈C0(P)be a function of Kozlov. Then the cocycle attractor associated to (10) is chaotic in the Li-Yorke sense. Proof. There exists p0∈Pfwith sup t∈RZt 0 h(θsp0)ds =ρ < ∞. If 0 ≤x0≤r0 ρthen x(t, p0;x0) = x0eZt 0 h(θsp0)ds .