Zero-Hopf bifurcation in a Chua system

Agraïments: The first author is supported by the FAPESP-BRAZIL grants 2010/18015-6, 2012/05635-1, and 2013/25828-1. The second author is partially supported by FEDER-UNAB-10-4E-378, and a CAPES grant 88881. 030454/2013-01 do Programa CSF-PVE.A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are ±ωi ̸= 0 and 0. In general for a such equilibrium there is no theory for knowing when it bifurcates some small-amplitude limit cycles moving the parameters of the system. Here we study the zero-Hopf bifurcation using the averaging theory. We apply this theory to a Chua system depending on 6 parameters, but the way followed for studying the zero-Hopf bifurcation can be applied to any other di erential system in dimension 3 or higher. In this paper rst we show that there are three 4-parameter families of Chua systems exhibiting a zero-Hopf equilibrium. After, by using the averaging theory, we provide su cient conditions for the bifurcation of limit cycles from these families of zero-Hopf equilibria. From one family we can prove that 1 limit cycle bifurcates, and from the other two families we can prove that 1, 2 or 3 limit cycles bifurcate simultaneously