Zero-Hopf Bifurcations in Three-Dimensional Chaotic Systems with One Stable Equilibrium

Abstract

In [Molaie et al., 2013] the authors provided the expressions of 23 quadratic differential systems in R3 with the unusual feature of having chaotic dynamics coexisting with one stable equilibrium point. In this paper, we consider 23 classes of quadratic differential systems in R3 depending on a real parameter a, which, for a = 1, coincide with the differential systems given by [Molaie et al., 2013]. We study the dynamics and bifurcations of these classes of differential systems by varying the parameter value a. We prove that, for a = 0, all the 23 considered systems have a nonisolated zero-Hopf equilibrium point located at the origin. By using the averaging theory of first order, we prove that a zero-Hopf bifurcation takes place at this point for a = 0, which leads to the creation of three periodic orbits bifurcating from it for a > 0 small enough: an unstable one and a pair of saddle type periodic orbits, that is, periodic orbits with a stable and an unstable manifold. Furthermore, we numerically show that the hidden chaotic attractors which exist for these systems when a = 1 are obtained by period-doubling route to chaos.