Inversely Unstable Solutions of Two-Dimensional Systems on Genus-p Surfaces and the Topology of Knotted Attractors

TL;DR

This paper investigates the complex dynamics of nonlinear systems on genus-p surfaces, revealing conditions for chaotic attractors through the study of inversely unstable solutions.

Contribution

It generalizes previous results to higher-genus surfaces and establishes criteria for non-circular invariant sets indicating chaos.

Findings

Existence of invariant attractors on genus-p surfaces.

Conditions for invariant sets to be non-homeomorphic to circles.

Identification of inversely unstable solutions as indicators of chaos.

Abstract

In this paper, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by studying the appearance of inversely unstable solutions within each invariant set.