Topology and dynamics of a flow that has a non-saddle set or a $W$-set

TL;DR

This paper investigates the topological and dynamical properties of flows near non-saddle and W-sets, providing classifications, cohomological relations, and insights into robustness and bifurcations in 2-manifolds.

Contribution

It offers a dynamical classification of surfaces based on non-saddle and W-sets, and analyzes their robustness and bifurcation behaviors in 2-manifolds.

Findings

Established cohomological relations between non-saddle sets and manifolds.

Provided a classification of surface dynamics involving non-saddle and W-sets.

Analyzed robustness and bifurcation properties of these sets in 2-manifolds.

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 or a $W$-set in a manifold. These are certain classes of compact invariant sets in whose vicinity the asymptotic behaviour of the flow is somewhat controlled. We are mainly concerned with global properties of the dynamics and establish cohomological relations between the non-saddle set and the manifold. As a consequence we obtain a dynamical classification of surfaces (orientable and non-orientable). We also examine robustness and bifurcation properties of non-saddle-sets and study in detail the behavior of $W$-sets in 2-manifolds.