Dissipative flows, global attractors and shape theory

TL;DR

This paper investigates the stability and continuation of global attractors in dissipative flows, employing shape theory and Conley index to analyze bifurcations from global to non-global attractors.

Contribution

It provides a necessary and sufficient condition for the persistence of global attractors under perturbations, advancing understanding of bifurcations in dissipative dynamical systems.

Findings

Characterization of conditions for global attractor continuation

Application of shape theory and Conley index to bifurcation analysis

Identification of criteria for global to non-global bifurcations

Abstract

In this paper we study continuous parametrized families of dissipative flows, which are those flows having a global attractor. The main motivation for this study comes from the observation that, in general, global attractors are not robust, in the sense that small perturbations of the flow can destroy their globality. We give a necessary and sufficient condition for a global attractor to be continued to a global attractor. We also study, using shape theoretical methods and the Conley index, the bifurcation global to non-global.