Periodic orbits in the zero-Hopf bifurcation of the Rössler system

A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are ±ωi ̸= 0 and 0. For a such equilibrium there is no a general theory for knowing when from this equilibrium bifurcates a small-amplitude periodic orbit moving the parameters of the system. We provide here an algorithm for solving this problem. In particular, first we characterize the values of the parameters for which a zero-Hopf equilibrium point takes place in the Rössler systems, and we find two one-parameter families exhibiting such equilibria. After for one of these families we prove the existence of one periodic orbit bifurcating from the zero-Hopf equilibrium. The algorithm developed for studying the zero-Hopf bifurcation of the Rössler systems can be applied to other differential system in Rn.