Degenerate Hopf bifurcations in a family of FF-type switching systems

The theory of Hopf bifurcation for switching systems is different from that for smooth systems because the nonsmoothness may produce more limit cycles. Although some results were obtained and several versions of order were given for switching weak foci, few works were found for switching centers or the general relationship between order and cyclicity. In this paper we study degenerate Hopf bifurcations in a family of FF-type switching systems, whose unperturbed upper system and lower one both have the same equilibrium of linear center type on the sole switching line. We use fractional orders to uniform the concept of order of weak foci for both the smooth case and the switching one and generally discuss the relationship between the order of weak foci and the cyclicity and give the uniform upper bound for the cyclicity in terms of the Bautin depth of the ideal generated by the switching Lyapunov quantities. Moreover, we investigate the independence of those switching Lyapunov quantities. We finally apply our results to a family of switching quadratic Bautin systems and prove that the cyclicity is at least 5 and 8 in the cases of weak foci and centers respectively.