Conley index of Poincaré maps in isolating segments

We calculate the discrete-time Conley index of the Poincaré map of a time-periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment. As an application, we present results on the existence of bounded solutions of some planar equations.     

Polygonal approximation of flows

The analysis of the qualitative behavior of flows generated by ordinary differential equations often requires quantitative information beyond numerical simulation which can be difficult to obtain analytically. In this paper we present a computational scheme designed to capture qualitative information using ideas from the Conley index theory. Specifically we design an combinatorial multivalued approximation from a simplicial decomposition of the phase space, which can be used to extract isolating blocks for isolated invariant sets. These isolating blocks can be computed rigorously to provide computer-assisted proofs. We also obtain local conditions on the underlying simplicial approximation that guarantees that the chain recurrent set can be well-approximated.     

The external multiplication for the Conley index

We construct an additional operation of the external multiplication on the cohomological Conley index defined by Mrozek for discrete semidynamical systems. The construction is based on the notion of the Conley index over a phase space introduced by Szybowski. We show how to apply the external multiplication to solve the problem of continuation of two isolated invariant sets and illustrate it by an example.     

The Conley Index on Compact Anr’s Is of Finite Type

We show that isolated invariant sets of a flow coincide with isolated invariant sets of the time-one-map of the flow. We prove that the cohomological Conley index of an isolated invariant set of a homeomorphism in a compact metric ANR has only finite number of non-zero ingredients and all of them are finite dimensional. As a consequence we obtain the same result for flows.     

Realization of all Dold's congruences with stability

The main goal of this paper is to prove that for each n>2, every sequence of integers satisfying Dold's congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving Rn-homeomorphism at an isolated stable fixed point. We use Conley index techniques even though stable fixed points are not isolated invariant sets.     

Conley index condition for asymptotic stability

In this paper, we use Conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous-time systems. The Conley index is a topological generalization of the Morse theory which has been developed to analyze dynamical systems using topological methods. In particular, the Conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set. The Conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set, including system stability properties. Efficient numerical algorithms using homology theory have been developed in the literature to compute the Conley index and can be used to deduce the stability properties of nonlinear dynamical systems.     

Stability and extensibility results for abstract skew-product semiflows

In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed.     

The boundary map and the connecting orbits

The purpose of this article is to show that the image of the homological boundary map attached to a filtration for an attractor-repeller pair of a smooth flow on a compact manifold is a submodule of the Alexander cohomology of certain order of the connecting set (some restrictions have to be imposed in order to have a valid argument). In particular, this gives an affirmative answer to a conjecture in Conley index theory which states that if the boundary map is not zero in two dimensions, the connecting set cannot be contractible.     

Asymptotically finite dimensional pullback behaviour of non-autonomous PDEs

The asymptotic behaviour of general non-autonomous partial differential equations can be described using the concept of pullback attractor. This is, under suitable hypotheses, a time-dependent family of finite-dimensional compact sets. In this work we investigate how this finite-dimensional dynamics on the attractor determines the asymptotic behaviour of non-autonomous PDEs.     

Some properties for the attractors

For the continuous flows defined on a topological space, we have discussed some properties for the invariant sets and their domains of influence. We have proved the following open problem posed by C. Conley: an attractor in a locally connected compact Hausdorff invariant set has finitely many components. In the meantime, two necessary and sufficient conditions for a set to be an attractor have been given.     

The rotation class of a flow

Generalizing a construction of A. Weil, we introduce a topological invariant for flows on compact, connected, finite dimensional, Abelian, topological groups. We calculate this invariant for some examples.     

The existence and uniqueness of trajectories joining critical points for differential equations in R3

In this paper, we prove the existence and uniqueness of trajectories joining critical points for differential equations in R³ by constructing the index pair of the isolated invariant set and using Conley index theory.     

A non-autonomous Conley index

In this papier, a homotopy index (Conley index) which can be applied to non-autonomous differential equations is defined. It is proved that the index is well defined, and several theorems concerning its basic properties are established. The second part of this paper is concerned with the application of this index to (non-autonomous) ordinary differential equations as well as (non-autonomous) semilinear parabolic equations. Finally, several existence results for bounded solutions of asymptotically linear non-autonomous equations are proved. We also consider the existence of recurrent or Poisson stable solutions.     

Numerical determination of the basin of attraction for exponentially asymptotically autonomous dynamical systems

Numerical methods to determine the basin of attraction for autonomous equations focus on a bounded subset of the phase space. For non-autonomous systems, any relevant subset of the phase space, which now includes the time as one coordinate, is unbounded in the t-direction. Hence, a numerical method would have to use infinitely many points.To overcome this problem, we introduce a transformation of the phase space. Restricting ourselves to exponentially asymptotically autonomous systems, we can map the infinite time interval to a finite, compact one. The basin of attraction of a solution becomes the basin of attraction of an exponentially stable equilibrium for an autonomous system. Now we are able to generalise numerical methods from the autonomous case. More precisely, we characterise a Lyapunov function as a solution of a suitable linear first-order partial differential equation and approximate it using radial basis functions.     

An approximation scheme for defining the Conley index of isolated critical points

In the present paper, we study isolated critical points of functionals dened on a real separable Hilbert space H and satisfying the H-properness condition. We introduce the notion of Conley index of an isolated critical point and prove that it is homotopy invariant. The scheme suggested here for defining the Conley index is based on the application of finite-dimensional Conley index theory to finite-dimensional restrictions of the functional to be studied.Translated from Differentsialnye Uravneniya, Vol. 40, No. 11, 2004, pp. 1462–1467. Original Russian Text Copyright © 2004 by Bobylev, Bulatov, Kuznetsov.     

Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations

The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs).     

Rigorous Numerics for Partial Differential Equations: The Kuramoto—Sivashinsky Equation

We present a new topological method for the study of the dynamics of dissipative PDEs. The method is based on the concept of the self-consistent a priori bounds, which permit the rigorous justification of the use of Galerkin projections. As a result, we obtain a low-dimensional system of ODEs subject to rigorously controlled small perturbation from the neglected modes. To these ODEs we apply the Conley index to obtain information about the dynamics of the PDE under consideration. We applied the method to the Kuramoto—Sivashinsky equation We obtained a computer-assisted proof of the existence of several fixed points for various values of ν > 0 . May 10, 2000. Final version received: March 9, 2001.     

Chain prolongation and chain stability

In this article we introduce chain prolongation, with which we define the concept of chain stability that takes an intermediate position between absolute stability and asymptotic stability. Two characterizations of chain stability are given, in terms of a Lyapunov function and a fundamental system of neighborhoods. As a matter of fact, a positively invariant compact set is chain stable if and only if it is a quasi-attracting set.     

Equi-attraction and backward compactness of pullback attractors for point-dissipative Ginzburg-Landau equations

A new concept of an equi-attractor is introduced, and defined by the minimal compact set that attracts bounded sets uniformly in the past, for a non-autonomous dynamical system. It is shown that the compact equi-attraction implies the backward compactness of a pullback attractor. Also, an eventually equi-continuous and strongly bounded process has an equi-attractor if and only if it is strongly point dissipative and strongly asymptotically compact. Those results primely strengthen the known existence result of a backward bounded pullback attractor in the literature. Finally, the theoretical criteria are applied to prove the existence of both equi-attractor and backward compact attractor for a Ginzburg-Landau equation with some varying coefficients and a backward tempered external force.