Detecting the Shilnikov scenario in a Hopf-Hopf bifurcation with 1:3 resonance

We investigate the behaviour of the primary solutions at a Hopf-Hopf interaction close to a 1: 3 resonance. It turns out, that the secondary bifurcations from the primary periodic solution branches are governed by Duffing and Mathieu equations. By numerical path following a homoclinic orbit at a saddle node was detected, giving rise to the Shilnikov scenario. In order to understand the creation of homoclinic orbits, a continuation of that orbit was applied, which terminated at an equilibrium with a triple zero eigenvalue. The existence of different types of homoclinic and heteroclinic orbits in the vicinity of triple zero bifurcation points has already been established. A short discussion of the local bifurcations at the triple zero eigenvalue is given. (C) 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).