Ordered intricacy of Shilnikov saddle-focus homoclinics in symmetric systems

Abstract

Using the technique of Poincaré 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.

Homoclinic chaos in the Rössler model

“…As was shown in 37 , these secondary hubs lie inside the U-shaped curves h i corresponding to the primary homoclinic loops. Note that the distinctive U-shape of the bifurcation curves is a common feature of diverse applications with a homoclinic saddle-focus, see for example [44][45][46][47] and the references therein. In this subsection, we show how 1D maps help to predict the form of homoclinic loops originating from the most visible secondary hubs located inside the curve h 1 .…”