Ordered Intricacy Of Shilnikov Saddle-Focus Homoclinics In Symmetric Systems

Using the technique of Poincare return maps, we disclose an intricate order of subsequent homoclinic bifurcations near the primary figure-8 connection of the Shilnikov saddle-focus in systems with reflection symmetry. We also reveal admissible shapes of the corresponding bifurcation curves in a parameter space. Their scalability ratio and organization are proven to be universal for such homoclinic bifurcations of higher orders. Two applications with similar dynamics due to the Shilnikov saddle-foci are used to illustrate the theory: a smooth adaptation of the Chua circuit and a 3D normal form. Published under an exclusive license by AIP Publishing.