Resonances of Periodic orbits in RÖssler System in Presence of a Triple-Zero bifurcation

This paper focuses on resonance phenomena that occur in a vicinity of a linear degeneracy corresponding to a triple-zero eigenvalue of an equilibrium point in an autonomous tridimensional system. Namely, by means of blow-up techniques that relate the triple-zero bifurcation to the Kuramoto–Sivashinsky system, we characterize the resonances that appear near the triple-zero bifurcation. Using numerical tools, the results are applied to the Rossler equation, showing a number of interesting bifurcation behaviors associated to these resonance phenomena. In particular, the merging of the periodic orbits appeared in resonances, the existence of two types of Takens–Bogdanov bifurcations of periodic orbits and the presence of Feigenbaum cascades of these bifurcations, joined by invariant tori curves, are pointed out.