Saddle-point-Saddle-focus singular cycles and existence of horseshoes

TL;DR

This paper investigates a specific type of singular cycle in 3D vector fields, revealing a new mechanism for chaos creation through saddle-focus singular cycles with transverse heteroclinic intersections.

Contribution

It introduces a novel class of singular cycles involving saddle-focus points that lead to chaos, expanding understanding of chaotic dynamics in three-dimensional systems.

Findings

Identification of a new chaos-generating mechanism

Analysis of saddle-focus singular cycles with transverse heteroclinic orbits

Implications for the structure of chaotic attractors in 3D vector fields

Abstract

In this paper we study a type of two singular point singular cycle where one heteroclinic orbit is the transversal intersection of the 2-dimensional stable manifold of one singular point and the 2-dimensional unstable manifold of other singular point. We show that this kind of singular cycle can give rise to a new mechanism of creation of chaos in 3-dimensional vector fields.