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A. Steindl:
"Detecting the Shilnikov scenario in a Hopf-Hopf bifurcation with 1:3 resonance";
Talk: IUTAM Symposium Analytical Methods in Nonlinear Dynamics, Frankfurt am Main; 2015-07-06 - 2015-07-09; in: "IUTAM Symposium Analytical Methods in Nonlinear Dynamics", P. Hagedorn et al. (ed.); Procedia IUTAM/Elsevier, (2015), ISSN: 2210-9838; 1 - 8.

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.

Hopf-Hopf bifurcation; resonance; Shilnikov scenario; Duffing equation; Mathieu equation; Homoclinic orbit; triple zero eigenvalue

http://publik.tuwien.ac.at/files/PubDat_244533.pdf