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 [Formula: see text] 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 [Formula: see text] depending on a real parameter [Formula: see text], which, for [Formula: see text], 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 [Formula: see text]. We prove that, for [Formula: see text], 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 [Formula: see text], which leads to the creation of three periodic orbits bifurcating from it for [Formula: see text] 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 [Formula: see text] are obtained by period-doubling route to chaos.