Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

Examples of attracting sets of birkhoff type

We discuss an explicit example of a map of the plane R ² with a nontrivial attracting set. In particular, we are concerned with the concept of rotation number introduced by Poincaré for maps of the circle and its subsequent extension by Birkhoff to maps of the annulus. The use of rotation number allows nontrivial attractors to be distinguished. The map we discuss has an attracting set containing a set of orbits with infinitely many different rotation numbers. We obtain the map by considering an Euler iteration of a family of vector fields originally described by Arnold and find that the resulting Euler map undergoes some bifurcations which are analogous to those of the family of vector fields. Specifically, there are Hopf bifurcations where changes of stability of a fixed point result in the creation of an attracting circle. The circle which grows from the fixed point is then shown to undergo structural changes giving nontrivial attracting sets. This arises from Euler map behaviour for which the corresponding vector field behaviour is a heteroclinic saddle connection. It is possible to give an explicit trapping region for the Euler map which contains the attracting set and to describe some of its properties. Finally, an analogy is drawn with attracting sets which arise for forced oscillators.     

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.     

On the magnification of cantor sets and their limit models

For aC ¹⁺ hyperbolic (cookie-cutter) Cantor setC we consider the limits of sequences of closed subsets ofR obtained by arbitrarily high magnifications around different points ofC. It is shown that a well defined set of limit models exists for the infinitesimal geometry, orscenery, in the Cantor set. IfCC} is a diffeomorphic copy ofC then the set of limit models of C is the same as that ofC. Furthermore every limit model is made of Cantor sets which areC ¹⁺ diffeomorphic withC (for some >0, (0,1)), but not all suchC ¹⁺ copies ofC occur in the limit models. We show the relation between this approach to the asymptotic structure of a Cantor set and Sullivan's scaling function. An alternative definition of a fractal is discussed.With 1 Figure     

Local instability and localization of attractors. From stochastic generator to Chua's systems

This paper discusses the connection between various instability definitions (namely, Lyapunov instability, Poincaré or orbital instability, Zhukovskij instability) and chaotic movements. It is demonstrated that the notion of Zhukovskij instability is the most adequate for describing chaotic movements. In order to investigate this instability, a new type of linearization is offered and the connection between that and the theorems of Borg, Hartman-Olech, and Leonov is established. By means of new linearization, analytical conditions of the existence of strange attractors for impulse stochastic generators are obtained. The assumption is expressed that an analogous analytical tool may be elaborated for continuous dynamical systems describing Chua's circuits. The paper makes a first step in this direction and establishes a frequency criterion of the existence of positive invariant sets with positive Lebesgue measure for piecewise linear systems, which are unstable in every region of phase space where they are linear.     

A subdivision algorithm for the computation of unstable manifolds and global attractors

Summary. Each invariant set of a given dynamical system is part of the global attractor. Therefore the global attractor contains all the potentially interesting dynamics, and, in particular, it contains every (global) unstable manifold. For this reason it is of interest to have an algorithm which allows to approximate the global attractor numerically. In this article we develop such an algorithm using a subdivision technique. We prove convergence of this method in a very general setting, and, moreover, we describe the qualitative convergence behavior in the presence of a hyperbolic structure. The algorithm can successfully be applied to dynamical systems of moderate dimension, and we illustrate this fact by several numerical examples. Received May 11, 1995 / Revised version received December 6, 1995     

Tangent measure distributions of hyperbolic Cantor sets

Tangent measure distributions were introduced byBandt [2] andGraf [8] as a means to describe the local geometry of self-similar sets generated by iteration of contractive similitudes. In this paper we study the tangent measure distributions of hyperbolic Cantor sets generated by certain contractive mappings, which are not necessarily similitudes. We show that the tangent measure distributions of these sets equipped with either Hausdorff- or Gibbs measure are unique almost everywhere and give an explicit formula describing them as probability distributions on the set of limit models ofBedford andFisher [5].     

Lyapunov's direct method in the estimation of the Hausdorff dimension of attractors

This paper surveys results of the authors and others conceming estimates for the Hausdorff dimension of strange attractors, particularly in the case of (generalized) Lorenz systems and Rössler systems. A key idea is the interpretation of Hausdorff measure as an analogue of a Lyapunov function.     

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.     

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.     

A new proof of the stable manifold theorem

We give a new proof of the stable manifold theorem for hyperbolic fixed points of smooth maps. This proof shows that the local stable and unstable manifolds are projections of a relation obtained as a limit of the graphs of the iterates of the map. The same proof generalizes to the setting of stable and unstable manifolds for smooth relations.     

An analogue to the Smale—Birkhoff homoclinic theorem for iterated entire mappings

Summary A snapback repeller of an analytic mapping is defined as a full orbit which tends to an unstable fixed point backwards in time and snaps back to the same fixed point. This note gives a rather elementary proof that unstable periodic orbits accumulate near snapback repellers. The proof is entirely selfcontained and uses only standard elementary tools. We exploit that the global semiconjugacy of the entire analytic map to a linear map is itself an entire analytic function and apply the Theorem of Rouché to its zeros. We also generalize Marotto's result about the chaotic motion near a snapback repeller to include the degenerate case.     

On the behavior of attractors under finite difference approximation

We recall that the long-time behavior of the Kuramoto-Sivashinsky equation is the same as that of a certain finite system of ordinary differential equations. We show how a particular finite difference scheme approximating the Kuramoto-Sivashinsky may be viewed as a small C ¹ perturbation of this system for the grid spacing sufficiently small. As a consequence one may make deductions about how the global attractor and the flow on the attractor behaves under this approximation. For a sufficiently refined grid the long-time behavior of the solutions of the finite difference scheme is a function of the solutions at certain grid points, whose number and position remain fixed as the grid is refined. Though the results are worked out explicitly for the Kuramoto-Sivashinsky equation, the results extend to other infinite-dimensional dissipative systems.     

Graphs of groups and von Neumann dimension

The ℓ²-invariants of the fundamental group G of a graph of groups acting on a CW-complex X are related to the ℓ²-invariants of the edge and vertex groups of G acting on X. Various consequences are derived.     

Fixed point theorems of operators with ppf dependence in banach spaces

In this paper the theory of fixed point theorems of nonlinear whose domain and range are different Banach spaces are considered. Also the analogues of the Contraction Mapping Principle. Krasnoselskii's fixed point theorem and a result on the convergence of quasinonexpansive mappings are dealt with. As an application, the existence and uniqueness of solutions of differential equations with retarded argument in Banach spaces are discussed     

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.