Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems

TL;DR

This paper investigates the complex ordering of homoclinic orbits near a Shilnikov saddle-focus bifurcation in symmetric systems, revealing universal geometric patterns and bifurcation structures through theoretical analysis and applications.

Contribution

It introduces a detailed analysis of the ordered structure of homoclinic bifurcations in symmetric systems, including universal scaling laws and bifurcation curve shapes.

Findings

Disclosed intricate order of homoclinics near bifurcation

Identified universal shapes of bifurcation curves

Validated theory with applications to Chua circuit and 3D normal form

Abstract

Using the technique of Poincar\'{e} return maps, we disclose an intricate order of the subsequent homoclinics near the primary homoclinic bifurcation of the Shilnikov saddle-focus in systems with reflection symmetry. We also reveal the admissible shapes of the corresponding bifurcation curves in a parameter plane of such systems. The scalability ratio of geometry and organization is proven to be universal for such homoclinic bifurcations of higher orders. Two applications with similar dynamics due to the Shilnikov saddle-foci, a smooth adaptation of the Chua circuit and a 3D normal form, are used to illustrate the theory