Bifurcation structure of interval maps with orbits homoclinic to a saddle-focus

TL;DR

This paper investigates the bifurcation structure of an interval map with a saddle-focus, revealing how homoclinic bifurcations unfold in symmetric systems and providing a symbolic encoding approach for analyzing stability and bifurcation sets.

Contribution

It introduces a detailed analysis of homoclinic bifurcations in saddle-focus interval maps, including symbolic encoding and conditions for stability, advancing understanding of low-dimensional chaotic dynamics.

Findings

Identification of homoclinic bifurcation structures in the parameter space

Development of symbolic encoding for stability analysis

Insights into bifurcation curve shapes and symmetry effects

Abstract

We study homoclinic bifurcations in an interval map associated with a saddle-focus of (2, 1)-type in $\mathbb{Z}_2$-symmetric systems. Our study of this map reveals the homoclinic structure of the saddle-focus, with a bifurcation unfolding guided by the codimension-two Belyakov bifurcation. We consider three parameters of the map, corresponding to the saddle quantity, splitting parameter, and focal frequency of the smooth saddle-focus in a neighborhood of homoclinic bifurcations. We symbolically encode dynamics of the map in order to find stability windows and locate homoclinic bifurcation sets in a computationally efficient manner. The organization and possible shapes of homoclinic bifurcation curves in the parameter space are examined, taking into account the symmetry and discontinuity of the map. Sufficient conditions for stability and local symbolic constancy of the map are presented. This study furnishes insights into the structure of homoclinic bifurcations of the saddle-focus map, furthering comprehension of low-dimensional chaotic systems.