Dissonant points and the region of influence of non-saddle sets

TL;DR

This paper investigates the topological and dynamical properties of non-saddle sets in flows, focusing on dissonant points, their influence, and conditions for attractors or repellers, especially in 2D and on manifolds.

Contribution

It introduces new topological conditions for non-saddle sets to be attractors or repellers and analyzes dissonant points' role in flow complexity, including in 2D and on manifolds.

Findings

Dissonant points are key to understanding flow complexity.

Certain topological conditions guarantee non-saddle sets are attractors or repellers.

In planar flows, invariant continua with global influence are attractors or repellers.

Abstract

The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set. We see, in particular, that some topological conditions are sufficient to guarantee that these sets are attractors or repellers. We study in detail the existence of dissonant points of the flow, which play a key role in the description of the region of influence of a non-saddle set. These points are responsible for much of the dynamical and topological complexity of the system. We also study non-saddle sets from the point of view of the Conley index theory and consider, among other things, the case of flows on manifolds with trivial first cohomology group. For flows on these manifolds, dynamical robustness is equivalent to topological robustness. We carry out a particular study of 2-dimensional flows and give a topological condition which detects the existence of dissonant points for flows on surfaces. We also prove that isolated invariant continua of planar flows with global region of influence are necessarily attractors or repellers.