Richness of the dynamics at a Shilnikov bifurcation

Abstract

[en] In this work, we study the dynamics exhibited in 3 − dimensional parametric continuous dynamical systems containing a homoclinic orbit to a saddle-focus equilibrium. This setting gives rise to the Shilnikov bifurcation, which can be studied using an appropriate Poincaré section that reduces the original system into a discrete 2 − dimensional one. The bifurcation presents various cases, each showing rich and different dynamics. The Shilnikov Theorem describes one of the possible scenarios. This case follows from a careful analysis of a suitable return map that shows that dynamics in some regions is equivalent to the one of the horseshoe map. To illustrate properties and scenarios appearing at the bifurcation, we derive a family of systems with the desired properties and investigate them numerically.