Takens-Bogdanov bifurcations of periodic orbits and Arnold's Tongues in a Three-Dimensional Electronic Model

In this paper we study Arnold’s tongues in a Z2-symmetric electronic circuit. They appear in a rich bifurcation scenario organized by a degenerate codimension-three Hopf–pitchfork bifurcation. On the one hand, we describe the transition open-to-closed of the resonance zones, finding two different types of Takens–Bogdanov bifurcations (quadratic and cubic homoclinic-type) of periodic orbits. The existence of cascades of the cubic Takens–Bogdanov bifurcations is also pointed out. On the other hand, we study the dynamics inside the tongues showing different Poincaré sections. Several bifurcation diagrams show the presence of cusps of periodic orbits and homoclinic bifurcations. We show the relation that exists between two codimension-two bifurcations of equilibria, Takens–Bogdanov and Hopf–pitchfork, via homoclinic connections, period-doubling and quasiperiodic motions.