Compact Invariant Set Research Articles

Dissipation and Compact Attractors

We present an approach to the study of the qualitative theory of infinite dimensional dynamical systems. In finite dimensions, most of the success has been with the discussion of dynamics on sets which are invariant and compact. In the infinite dimensional case, the appropriate setting is to consider the dynamics on the maximal compact invariant set. In dissipative systems, this corresponds to the compact global attractor. Most of the time is devoted to necessary and sufficient conditons for the existence of the compact global attractor. Several important applications are given as well as important results on the qualitative properties of the flow on the attractor.

Asymptotic behavior of stochastic discrete complex Ginzburg–Landau equations

The complex Ginzburg–Landau equation is one of the most-studied equations in applied mathematics. We consider the discretization of complex Ginzburg–Landau equations on one dimensional lattice driven by a general Gaussian random field including the translation invariant one. The long time behavior of the sample paths and the distributions of solutions are studied respectively. Under the gauge nonlinear interaction, the dynamical behavior for the sample paths of the system is described by a global random attractor which is a random compact invariant set in a weighted Hilbert space. Furthermore the distributions of the system exponentially converge to the unique invariant measure of the system, that is the system is ergodic. The asymptotic compactness and dissipative method are important in our approach.

Localization of periodic orbits of the Rössler system under variation of its parameters

The localization problem of compact invariant sets of the Rossler system is considered in this paper. The main interest is attracted to a localization of periodic orbits. We establish a number of algebraic conditions imposed on parameters under which the Rossler system has no compact invariant sets contained in half-spaces z > 0; z < 0 and in some others. We prove that if parameters (a, b, c) of the Rossler system are such that this system has no equilibrium points then it has no periodic orbits as well. In addition, we give localization conditions of compact invariant sets by using linear functions and one quadratic function.

Suspension Flows Over Countable Markov Shifts

We introduce the notion of topological pressure for suspension flows over countable Markov shifts, and we develop the associated thermodynamic formalism. In particular, we establish a variational principle for the topological pressure, and an approximation property in terms of the pressure on compact invariant sets. As an application we present a multifractal analysis for the entropy spectrum of Birkhoff averages for suspension flows over countable Markov shifts. The domain of the spectrum may be unbounded and the spectrum may not be analytic. We provide explicit examples where this happens. We also discuss the existence of full measures on the level sets of the multifractal decomposition.

The Brouwer Fixed Point Theorem applied to rumour transmission

The results of the Brouwer Fixed Point Theorem are extended to continuous dynamical systems. It is shown that if there exists a compact convex positive invariant set for the dynamical system, then this convex positive invariant set contains an equilibrium point. The existence of an interior equilibrium is shown for a general model of rumour transmission.

ATTRACTORS FOR STOCHASTIC LATTICE DYNAMICAL SYSTEMS

We consider a one-dimensional lattice with diffusive nearest neighbor interaction, a dissipative nonlinear reaction term and additive independent white noise at each node. We prove the existence of a compact global random attractor within the set of tempered random bounded sets. An interesting feature of this is that, even though the spatial domain is unbounded and the solution operator is not smoothing or compact, pulled back bounded sets of initial data converge under the forward flow to a random compact invariant set.

Localization of compact invariant sets of the Lorenz system

The problem of finding domains in the state space of a nonlinear system which contain all compact invariant sets is considered. Such domains are computed for the Lorenz system by using different localizing functions.

INVARIANT SUBSETS OF VOLUME HYPERBOLIC SETS

A volume hyperbolic set is a compact invariant set with a dominated splitting whose external bundles uniformly contract and expand the volume respectively [1]. Examples of volume hyperbolic sets for diffeomorphisms or flows are the hyperbolic sets, the geometric Lorenz attractor [3] and the singular horseshoe [6]. We shall prove that no invariant subset of a volume hyperbolic set of a three-dimensional flow is homeomorphic to a closed surface.

Necessary and sufficient conditions for semi-uniform ergodic theorems and their applications

It has been established one-side uniformconvergence in both the Birkhoff and sub-additive ergodic theoremsunder conditions on growth rates with respect to all the invariantmeasures. In this paper we show these conditions are bothnecessary and sufficient. These results are applied to studyquasiperiodically forced systems. Some meaningful geometricproperties of invariant sets of such systems are presented. Wealso show that any strange compact invariant set of a$\mathcal{C}^1$ quasiperiodically forced system must support aninvariant measure with a non-negative normal Lyapunov exponent.

Localization of Invariant Compact Sets of Dynamical Systems

Localization of Invariant Compact Sets of Dynamical Systems

On the Structure of a Neighborhood of Stable Compact Invariant Sets

The analysis of the phase portrait of trajectories in a neighborhood of stable invariant sets of dynamical systems is one of the main problems in qualitative theory of stability of motion. This problem has been under study starting from Lyapunov’s work [1]; in particular, for abstract dynamical systems, it was considered in [2–8], where a wide bibliography on the topic can be found. However, only the case of asymptotic stability has been sufficiently well analyzed (see [1–3, 9]). The so-far existing practice of scientific research shows that, in the case of nonasymptotic stability, the qualitative picture of trajectories near compact invariant sets is very complicated and immensely variegated, which explains why the problem of qualitatively distinguishing all possible situations has not been satisfactorily solved yet. In the present paper, we suggest an approach to the study of the structure of a neighborhood of compact invariant sets for locally compact dynamical systems and use it to give some classification of types of orbitally nonasymptotically stable invariant sets. The analysis carried out below permits one to single out five basic classes of first-level sets and 121 subclasses of second-level sets. Related illustrative examples of dynamical systems are given. Let the triple (X, , π) denote the dynamical system defined on a locally compact metric space (X, d) with phase mapping π : X × → X (π : (x, t) → xt) [2, 3]. For each point x ∈ X, the mapping x : t → xt is referred to as the motion passing through x, and the sets

Dynamical behavior for stochastic lattice systems

Abstract Random attractors describe the long term behavior of the random dynamical systems. This paper is devoted to a general first order stochastic lattice dynamical systems (SLDS) with some dissipative nonlinearity. We prove the asymptotic compactness of the random dynamical system and obtain the random attractor, which is a compact random invariant set with tempered bound.

Dimension estimates beyond conformal and hyperbolic dynamics

We derive upper bounds for the upper box dimension of a compact invariant set in terms of the topological pressure of the exponential growth rate of k-volumes. In particular, for our approach the dynamics needs neither to be conformal nor uniformly hyperbolic.

Robustly expansive homoclinic classes

Let $f: M \to M$ be a diffeomorphism defined in a three-dimensional compact boundary-less manifold M. We prove that for an open dense set, C1-robustly expansive homoclinic classes H(p) for f are hyperbolic. A diffeomorphism f is $\alpha$-expansive on a compact invariant set K if there is $\alpha>0$ such that for all $x,y\in K$, if ${\rm dist}(f^n(x),f^n(y))\leq \alpha$ for all $n\in \mathbb Z$ then x = y. By ‘robustly’ we mean that there is $\alpha>0$ such that for all nearby diffeomorphisms g, the homoclinic class H(pg) of the continuation of p is $\alpha$-expansive.

An Elementary Proof for Hyperbolicity and Chaos of the Logistic Maps

It is well known, that for any the maximal compact invariant set of the logistic map is hyperbolic and that is chaotic on Λμ. While this result can be found in most books dealing with chaotic dynamics, its proof is usually given only for In this paper, we present an elementary and self-contained proof for all

A minimum principle for Lyapunov exponents and a higher-dimensional version of a theorem of Mañé

We consider compact invariant sets Λ forC1 maps in arbitrary dimension. We prove that if Λ contains no critical points then there exists an invariant probability measure with a Lyapunov exponent λ which is theminimum of all Lyapunov exponents for all invariant measures supported on Λ. We apply this result to prove that Λ isuniformly expanding if every invariant probability measure supported on Λ is hyperbolic repelling. This generalizes a well known theorem of Mane to the higher-dimensional setting.

Averaging, Conley index continuation and recurrent dynamics in almost-periodic parabolic equations

We study a non-autonomous parabolic equation with almost-periodic, rapidly oscillating principal part and nonlinear interactions. We associate to the equation a skew-product semiflow and, for a special class of nonlinearities, we define the Conley index of isolated compact invariant sets. As the frequency of the oscillations tends to infinity, we prove that every isolated compact invariant set of the averaged autonomous equation can be continued to an isolated compact invariant set of the skew-product semiflow associated to the non-autonomous equation. Finally, we illustrate some examples in which the Conley index can be explicitly computed and can be exploited to detect the existence of recurrent dynamics in the equation.

Asymptotics of parabolic equations with possible blow-up

We describe the long-time behaviour of solutions of parabolic equations in the case when some solutions may blow up in a finite or infinite time. This is done by providing a maximal compact invariant set attracting any initial data for which the correspon

Global Stability of Approximation for Exponential Attractors

This paper is concerned with the initial value problem for some diffusion system which describes the process of a pattern formation of biological individuals by chemotaixis and growth. In the paper Osaki et al. [13], exponential attractors have been constructed for the dynamical system determined by this problem. The exponential attractor is one of limit sets which is a positively invariant compact set with finite fractal dimension and which attracts every trajectory in an exponential rate. In this paper we study another feature of exponential attractors, that is we show that the approximate solution also gets close to the exponential attractor in an exponential rate and remains in its neighborhood forever. Our methods are available to any other exponential attractors determined by interaction-diffusion systems.

Transitive sets for quasi-periodically forced monotone maps

There has been considerable interest in recent years in quasi-periodically forced systems, partly due to the fact that these commonly exhibit strange non-chaotic attractors. Relatively little is known rigorously about such systems. In this paper we concentrate on investigating the structure of the simplest possible invariant sets for a particular class of quasi-periodically forced maps, namely those that are monotone in each fibre. Due to the quasi-periodic nature of the forcing, periodic orbits cannot occur, and their role is played by various types of invariant graph. Any compact invariant set is bounded by two invariant graphs, which are respectively upper and lower semi-continuous. If the set is minimal, these two graphs intersect on a residual set, on which both are continuous. Any transitive set Ω contains either one or two minimal sets, which must be the closure of one or the other of the boundaries of Ω. If Ω contains only one minimal set, then again its upper and lower boundaries intersect on a residual set. This case contains the original example of Grebogi et al. and its generalizations by Keller and by Glendinning. If Ω contains two minimal sets, then its upper and lower boundaries cannot intersect, though as far as we are aware, there is no known example where the existence of such a transitive set has been proven rigorously.