Bifurcations, robustness and shape theory of attractors of discrete dynamical systems

TL;DR

This paper explores the topological properties and bifurcations of attractors in discrete dynamical systems, focusing on robustness, shape theory, and relations between flows and homeomorphisms.

Contribution

It introduces new robustness results for attractors of plane homeomorphisms and connects attractors of flows with quasi-attractors in Rn, also examining shape properties of invariant sets.

Findings

Robustness properties of attractors under Andronov-Hopf bifurcation

Relations between flow attractors and homeomorphism quasi-attractors

Shape analysis of invariant sets in IFSs on the plane

Abstract

We study in this paper global properties, mainly of topological nature, of attractors of discrete dynamical systems. We consider the Andronov-Hopf bifurcation for homeomorphisms of the plane and establish some robustness properties for attractors of such homeomorphisms. We also give relations between attractors of flows and quasi-attractors of homeomorphisms in Rn. Finally, we give a result on the shape (in the sense of Borsuk) of invariant sets of IFSs on the plane, and make some remarks about the recent theory of Conley attractors for IFS.