Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems

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