Topological bifurcations of minimal invariant sets for set-valued dynamical systems

TL;DR

This paper investigates how minimal invariant sets in set-valued dynamical systems change with parameters, revealing that topological bifurcations are discontinuous and characterized by Morse-like decompositions, especially in systems with noise or control.

Contribution

It establishes the discontinuous nature of topological bifurcations in minimal invariant sets and characterizes these transitions using Morse-like decompositions.

Findings

Bifurcations are lower semi-continuous explosions or instantaneous appearances.

Discontinuities are with respect to the Hausdorff metric.

Results apply to random and control dynamical systems.

Abstract

We discuss the dependence of set-valued dynamical systems on parameters. Under mild assumptions which are often satisfied for random dynamical systems with bounded noise and control systems, we establish the fact that topological bifurcations of minimal invariant sets are discontinuous with respect to the Hausdorff metric, taking the form of lower semi-continuous explosions and instantaneous appearances. We also characterise these transitions by properties of Morse-like decompositions.