Some Problems In Random Dynamical Systems

One important aspect of the qualitative analysis of differential equations and dynamical systems is the study of asymptotic, long-term behavior of solutions/orbits. Hence much of dynamical systems involves the study of the existence and structure of invariant sets. The Conley index theory (see [30, 33, 139]), developed by Conley and his students, has been a very powerful tool in the study of dynamical systems, differential equations and bifurcation theory. Inertial manifolds play a very important role in describing asymptotic behavior of infinite dimensional dissipative dynamical systems.Random dynamical systems ("RDS" in short) arise in the modeling of many phenomena in physics, biology, economics, climatology, etc and the random effects often reflect intrinsic properties of these phenomena rather . than just to compensate for the defects in deterministic models. The history of study of random dynamical systems goes back to Ulam and von Neumann [152] and it has flourished since the 1980s due to the discovery that the solutions of stochastic ordinary differential equations yield a cocycle over a metric dynamical system which models randomness, i.e. a random dynamical system.In this thesis, we make attempt to generalize the Conley index theory to RDS, and to investigate the stochastic inertial manifolds of damped wave equations. In Chapter 2, we study Morse decomposition theory for both finite and infinite dimensional RDS, which gives the positive answer to an open problem put forward by Caraballo, Langa [17]. In Chapter 3, by introducing the concept of backward orbits for random semiflow, we generalize Conley's fundamental theorem of dynamical systems to both finite and infinite dimensional RDS. In Chapter 4, by introducing the concepts of random isolated invariant set, random shift equivalence and random homotopy etc, we are able to define Conley index for time discrete RDS. In the last chapter, we consider the asymptotic behavior of stochastic wave equations. We obtain the existence and the upper semi-continuity of stochastic inertial manifolds of stochastic wave equations. Now let us introduce the main results of the thesis.Morse decomposition theory for RDSFirstly we consider the Morse decomposition theory for random flow on compact metric spaces. For the random attractor-repeller pair, a special case of Morse decomposition, we obtain the following result.Theorem 1 Assumeφis an RDS on a compact metric space X and A, R are two disjoint invariant random compact sets. Then (A, R) is an attractor-repeller pair with strong fundamental neighborhood if and only if there exists a Lyapunov function L: such that:(i) is measurable for each , and is continuous for each ;(ii) L(ω,x) = 0 when , and L(ω,x) = 1 when ; (iii) forBy the above theorem we can obtain the Morse decomposition theorem for random flow:Theorem 2 Assumeφis an RDS on a compact metric space X and letD = {M₁, M₂,…, M_n} be a finite collection of mutually disjoint invariant random compact sets. Then D is a Morse decomposition forφon X with each A_i having a strong fundamental neighborhood if and only if there exists a Lyapunov function L : [0,1] such that: (i) L(ω, x) is measurable for each x∈X, and x L(ω, x) is continuous for each ;(ii) L is constant on each M_i, i.e. for andα_i is independent ofω, i = 1,..., n;(iii)(iv) forBy introducing the concept of backward orbits for RDS, we obtain Morse decomposition theorem for infinite dimensional RDS, which gives the positive answer to an open problem put forward by Caraballo, Langa [17]. Theorem 3 Assumeφis a random semiflow on the Polish space X, and S is an invariant random compact ofφ. Assume D =- {M_i}_iⁿ=1 is a Morse decomposition of S, determined by attractor-repeller pairs (A_i, R_i), i = 1,..., n. Then M_D determines the limiting behavior ofφon S. Moreover, there are no "cycles" between the Morse sets. More precisely, we have the following holds:(i) For any random variable x in S, there exists an entire orbit a through x such that and almost surely.(ii) Ifσis an entire orbit through the random variable x satisfying that M_p almost surely and M_q almost surely for some 1≤p, q≤n, then p≤q; Moreover, p = q if and only if a lies on M_p.(iii) Ifσ₁,... ,σ_l are l entire orbits through the random varibales x₁,...,x_l respectively such that for some 1≤j₀,...,j_l≤n, and for k = 1,... ,l, then j₀≤j_l. Moreover, j₀ < j_l if and only ifσ_k does not lie on M_D with positive probability for some k, otherwiseConley decomposition theorem for RDSNext let us consider the random case of Conley's fundamental theorem of dynamical systems. To do this, firstly we introduce the concept of random chain recurrence. Definition 1 For given random variable∈(ω) > 0, the n+1 pairs (x₀(ω),t₀), (x₁(ω),t₁),…, (x_n(ω),t_n), where x₀(ω), x₁(ω),…, x_n(ω) are random variables, are called a random∈(ω)-T(ω) -chain, if the following holds:where t_i≥T(ω), T(ω) is a positive random variable almost surely. And we call n the length of the random∈(ω)-T(ω)-chain. A random variable x(ω) is called random chain recurrent if for any given∈(ω),T(ω) > 0, there exists an∈(ω)-T(ω)-chain beginning and ending at x(ω) P-almost surely; x(ω) is called partly random chain recurrent with index 6 if for any∈(ω),T(ω) > 0, there exists an∈(ω)-T(ω) -chain beginning and ending at x(ω) with probability not less than 5, where the index 5 is the maximal number satisfying this property; x(ω) is called completely random non-chain recurrent if there exists∈₀(ω), T₀(ω) > 0 such that there is no∈₀(ω)-T₀(ω)-chain beginning and ending at x(ω) with positive probability.We denote CR_φ(ω) the random chain recurrent set ofφ, which has the property that for any random chain recurrent variable x(ω), we have x(ω)∈CR_φ(ω) P-almost surely, and vice versa (i.e. if a random variable x(ω)∈CR_φ(ω) P-almost surely, then x(ω) is random chain recurrent); for any completely random non-chain recurrent variable x(ω), we have x(ω)∈X - CR_φ(ω) P-almost surely, and vice versa; for any partly random chain recurrent variable x(ω) with index 5, we have x(ω)∈CR_φ(ω) with probabilityδ, and vice versa. For any given random variable x(ω), denoteIf x(ω) is a partly random chain recurrent variable with indexδ, then we call the chain recurrent part of x(ω). Therefore by the property of CR_φ(ω), we have that CR_φ(ω) is the union of all random chain recurrent variables and the chain recurrent part of those partly random chain recurrent variables. Then we obtain the following relation between random chain recurrent set and random attractors.Theorem 4 Assume X is a compact metric space, A(ω) is a random local attractor and R(ω) is the random repeller corresponding to A(ω), thenalmost surely, where the intersection is taken over all random local attractors.Now we give the definition of complete Lyapunov functions for RDS. Definition 2 A complete Lyapunov function for the RDS ip is an F×B(X)-measurable function L : with the following properties:(1) ;(2) .(3) The range of L on CR_φ(ω) is a compact nowhere dense subset of [0,1];(4) L separates different random chain transitive components ofφ. Theorem 5 (complete Lyapunov function). The function defined by (3.2.4) is a complete Lyapunov function for the RDSφ. Moreover, if C and C'are distinct random chain transitive components ofφwith the property that for arbitrary∈(ω),T(ω) > 0 there is an∈(ω)-T(ω)-chain from C to C P-a.s.,then L(Ω,C)>L(Ω,C').By Theorems 4 and 5, we obtain the random case of Conley's fundamental theorem of dynamical systems.Theorem 6 (random Conley decomposition theorem). Any random dynamical system (with a separable metric space endowed with a probability measure as base space) on a compact metric space decomposes the space into a random chain recurrent part and a random gradient-like part.The above result can be further extended to RDS on general Polish spaces: Theorem 7 (random Conley decomposition theorem on Polish spaces). Any random dynamical system (with a separable metric space endowed with a probability measure as base space) on a Polish space decomposes the Polish space into a random chain recurrent part and a random gradient-like part.We also prove that Conley decomposition theorem holds for random semiflow on Polish spaces:Theorem 8 Assume X is a Polish space andφis a random semiflow on X. Then there exists a complete Lyapunov function L : forφwith the following properties: (i) L is an F×B(X)-measurable function;(ii) , whereμis a positive measure onΩ×X;(iii) when ; (iv) If the random variable x is completely random non-chain recurrent, i.e. x(ω) P-a.s., then for arbitrary∈> 0 there exists an satisfying P such that for arbitrary ; the following holds:(v) The range of L on CR_φ(ω) is a compact nowhere dense subset of [0.1]; (vi) L separates different random chain transitive components ofφ; (vii) If C and C'are distinct random chain transitive components ofφwith the property that for arbitrary random variables∈, T > 0 there is an∈-T-chain from C to C'P-a.s. then L(Ω, C) > L(Ω, C').Random Conley indexFirstly we give the definitions of random isolating neighborhood, random isolated invariant set, exit set and random filtration pair. Definition 3 A random compact set N is called a random isolating neighborhood if it satisfies whereWe also simply write Inv(N,φ) as InvN when we need not specifyφor it is clear from context, which does not cause confusion. Correspondingly we call S a random isolated invariant set if there exists a random isolating neighborhood N such that S = Inv(N,φ). A random compact set N is called a random isolating block if for P-almost allω, it satisfiesDefinition 4 For a random set N we define the exit set of N, denoted by N^-, to beDefinition 5 Assume N is a random isolating neighborhood, L (?) N is a random compact set and S is the random isolated invariant set inside N. We also assume N = cl(intJV), L = cl(intL). We call (N,L) is a random. filtration pair for 5 if the following holds:Now we can give the definition of Conley index for time discrete RDS as follows:Definition 6 Assumeφis the time one map of a discrete random dynamical system, S is a random isolated invariant set forφand P = (N,L) is a random filtration pair for S. Denote hp(S,φ) the random homotopy class [φp] on the random pointed space N_L withφP a representative element, then we define the random Conley index h(S,φ) for S to be the random shift equivalent class of hp(S,φ).Similar to the deterministic case, the random Conley index has continuation property and Wazewski property.Theorem 9 (Continuation property). Assumeφλ,λ∈[0,1] is a family of random homeomorphisms, which depends continuously (in the random C⁰ topology) onλ. If N is a random isolating neighborhood for eachφλ,λ∈[0,1], then the random Conley index h(S_λ,φλ) forφλis independent of [0,1], i.e. h(S_λ,φλ) = h(S₀,φ₀), where S_λ:= Inv(N,φλ),λ∈[0,1] is the random isolated invariant set forφλin N.Theorem 10 (Wazewski property). Assume S is a random isolated invariant set and the random Conley index for S is not trivial, i.e. h(S,λ)≠0, then 5≠(?) almost surely whenθis ergodic under P.Stochastic inertial manifoldsFirstly let us recall the definition of stochastic inertial manifolds. Definition 7 A random set is called invariant for RDSφif, for any t≥0. If an invariant set M(ω) can be represented by a Lipschitz or C^k mappingsuch thatthen we call M(ω) a Lipschitz or C^h invariant manifold. Furthermore, if PH is finite dimensional and M(ω) attracts exponentially all the orbits ofφ, then we call M(ω) a stochastic inertial manifold ofφ.Consider the following wave equation in [0,Ï€] perturbed by additive white noise: withwhere , . We assume that the nonlinear term f is globally Lipschitz continuous on L²(0,Ï€) with Lipschitz constant Lip f. We obtain the following two theorems.Theorem 11 Consider stochastic wave equation (1). There exists some such that for any , the equation (1) has a stochastic IM. Theorem 12 Assume M_δ(ω) is a stochastic IM of (1) and M₀ is the IM of (1) whenδ= 0 with the same dimension as that of M_δ(ω), then, for any R > 0, we havealmost surely.