Attractors and orbit-flip homoclinic orbits for star flows

TL;DR

This paper investigates star flows on closed 3-manifolds, establishing conditions under which they have finitely many attractors or can be approximated by vector fields with orbit-flip homoclinic orbits, advancing understanding of their dynamical complexity.

Contribution

It provides a dichotomy for star flows on 3-manifolds, linking the finiteness of attractors to the presence of orbit-flip homoclinic orbits in approximations.

Findings

Star flows have either finitely many attractors or can be approximated by vector fields with orbit-flip homoclinic orbits.

The study characterizes the structure of star flows on closed 3-manifolds.

It advances the classification of dynamical behaviors in 3-dimensional flows.

Abstract

We study star flows on closed 3-manifolds and prove that they either have a finite number of attractors or can be $C^1$ approximated by vector fields with orbit-flip homoclinic orbits.