Sufficient conditions for the existence of global random attractors for stochastic lattice dynamical systems and applications

In this paper, we first present some sufficient conditions for the existence of a global random attractor for general stochastic lattice dynamical systems. These sufficient conditions provide a convenient approach to obtain an upper bound of Kolmogorov ε-entropy for the global random attractor. Then we apply the abstract result to the stochastic lattice sine-Gordon equation.     

Random attractors for quasi-continuous random dynamical systems and applications to stochastic reaction-diffusion equations

This paper first introduces the so-called quasi-continuous random dynamical system (RDS) on a separable Banach space. The quasi-continuity is weaker than all the usual continuities and thus is easier to check in practice. We then establish a necessary and sufficient condition for the existence of random attractors for the quasi-continuous RDS. We also give a general method to obtain the random attractors for the RDS on the Banach space Lq(D) for q?2. As an application, it is shown that the RDS generated by the stochastic reaction-diffusion equation possesses a finite-dimensional random attractor in Lq(D) for any q?2, a comparison result of fractal dimensions under the different Lq-norms is also obtained.     

Exponential attractors for first-order lattice dynamical systems

In l², we investigate the existence of an exponential attractor for the solution semigroup of a first-order lattice dynamical system acting on a closed bounded positively invariant set which needs not to be compact since l² is infinite dimensional. Up to our knowledge, this is the first time to examine the existence of exponential attractors for lattice dynamical systems.     

Random attractors for second-order stochastic lattice dynamical systems

In this paper, the asymptotic behavior of second-order stochastic lattice dynamical systems is considered. We firstly show the existence of an absorbing set. Then an estimate on tails of the solutions is derived when the time is large enough, which ensures the asymptotic compactness of the random dynamical system. Finally, the existence of the random attractor is provided.     

Morse equation of attractors for nonsmooth dynamical systems

This paper is concerned with a Morse theory of attractors for finite-dimensional nonsmooth dynamical systems described by differential inclusions with upper semi-continuous righthand sides. We first show that all open attractor neighborhoods of an attractor share the same homotopy type. Then based on this basic fact we introduce the concept of homology index for Morse sets and establish Morse inequalities and Morse equation by using smooth Morse–Lyapunov functions.     

Pullback attractors for asymptotically compact non-autonomous dynamical systems

First, we introduce the concept of pullback asymptotically compact non-autonomous dynamical system as an extension of the similar concept in the autonomous framework. Our definition is different from that of asymptotic compactness already used in the theory of random and non-autonomous dynamical systems (as developed by Crauel, Flandoli, Kloeden, Schmalfuss, amongst others) which means the existence of a (random or time-dependent) family of compact attracting sets. Next, we prove a result ensuring the existence of a pullback attractor for a non-autonomous dynamical system under the general assumptions of pullback asymptotic compactness and the existence of a pullback absorbing family of sets. This attractor is minimal and, in most practical applications, it is unique. Finally, we illustrate the theory with a 2D Navier–Stokes model in an unbounded domain.     

Attractors for random dynamical systems

Summary A criterion for existence of global random attractors for RDS is established. Existence of invariant Markov measures supported by the random attractor is proved. For SPDE this yields invariant measures for the associated Markov semigroup. The results are applied to reation diffusion equations with additive white noise and to Navier-Stokes equations with multiplicative and with additive white noise.     

Hierarchical structure of attractors of dynamical systems

Some basic properties of the small random perturbed dynamical system of Freidlin-Wentzell type are elicited. A hierarchy structure of attractors is constructed and some further properties of this structure are confirmed. Project supported by the National Natural Science Foundation of China and the Doctoral Program Foundation of IHE 863 Program, China.     

Sternberg theorems for random dynamical systems

In this paper, we prove the smooth conjugacy theorems of Sternberg type for random dynamical systems based on their Lyapunov exponents. We also present a stable and unstable manifold theorem with tempered estimates that are used to construct conjugacy. © 2005 Wiley Periodicals, Inc.     

Conley index for random dynamical systems

Conley index theory is a very powerful tool in the study of dynamical systems. In this paper, we generalize Conley index theory to discrete random dynamical systems. Our constructions are basically the random version of Franks and Richeson in [J. Franks, D. Richeson, Shift equivalence and the Conley index, Trans. Amer. Math. Soc. 352 (2000) 3305-3322] for maps, and the relations of isolated invariant sets between time-continuous random dynamical systems and corresponding time-h maps are discussed. Two examples are presented to illustrate results in this paper.     

Mean-square random dynamical systems

The classical theory of random dynamical systems is a pathwise theory based on a skew-product system consisting of a measure theoretic autonomous system that represents the driving noise and a topological cocycle mapping for the state evolution. This theory does not, however, apply to nonlocal dynamics such as when the dynamics of a sample path depends on other sample paths through an expectation or when the evolution of random sets depends on nonlocal properties such as the diameter of the sets. The authors showed recently in terms of stochastic morphological evolution equations that such nonlocal random dynamics can be characterized by a deterministic two-parameter process from the theory of nonautonomous dynamical systems acting on a state space of random variables or random sets with the mean-square topology. This observation is exploited here to provide a definition of mean-square random dynamical systems and their attractors. The main difficulty in applying the theory is the lack of useful characterizations of compact sets of mean-square random variables. It is illustrated through simple but instructive examples how this can be avoided in strictly contractive cases or circumvented by using weak compactness. The existence of a pullback attractor then follows from the much more easily determined mean-square ultimate boundedness of solutions.     

Continuous random dynamical systems

We study a dynamical system generalizing continuous iterated function systems and stochastic differential equations disturbed by Poisson noise. The main results provide us with sufficient conditions for the existence and uniqueness of an invariant measure for the considered system. Since the dynamical system is defined on an arbitrary Banach space (possibly infinite dimensional), to prove the existence of an invariant measure and its stability we make use of the lower bound technique developed by Lasota and Yorke and extended recently to infinite-dimensional spaces by Szarek. Finally, it is shown that many systems appearing in models of cell division or gene expressions are exactly as those we study. Hence we obtain their stability as well.     

Existence of backwards-compact pullback attractors for non-autonomous lattice dynamical systems

A pullback attractor is called backward compact if the union of attractors over the past time is pre-compact. We show that this kind of attractor exists for the first-order non-autonomous lattice dynamical system when the external force is backwards tempered and backwards asymptotically tail-null.     

Uniform exponential attractors for first order non-autonomous lattice dynamical systems

In Abdallah (2008, 2009) [2] and [3], we have investigated the existence of exponential attractors for first and second order autonomous lattice dynamical systems. Within this work, in l², we carefully study the existence of a uniform exponential attractor for the family of processes associated with an abstract family of first order non-autonomous lattice dynamical systems with quasiperiodic symbols acting on a closed bounded set.     

Random attractors for stochastic lattice dynamical systems in weighted spaces

In this paper, we first provide some sufficient conditions for the existence of global compact random attractors for general random dynamical systems in weighted space (p?1) of infinite sequences. Then we consider the existence of global compact random attractors in weighted space for stochastic lattice dynamical systems with random coupled coefficients and multiplicative/additive white noises. Our results recover many existing ones on the existence of global random attractors for stochastic lattice dynamical systems with multiplicative/additive white noises in regular l² space of infinite sequences.