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

摘要

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.