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 Z(2)-symmetric electronic circ uit. They appear in a rich bifurcation scenario organized by a degenerate c odimension-three Hopf-pitchfork bifurcation. On the one hand, we describe t he 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, vie study the dynamic s inside the tongues showing different Poincare sections. Several bifurcati on diagrams show the presence of cusps of periodic orbits and homoclinic bi furcations. We show the relation that exists between two codimension-two bi furcations of equilibria, Takens-Bogdanov and Hopf-pitchfork, via homoclini c connections, period-doubling. and quasiperiodic motions.