Zero-Hopf bifurcation in a Chua system

Abstract

A zero-Hopf equilibrium is an isolated equilibrium point whose eigenvalues are $\pm \omega i\neq 0$ and $0$. In general for a such equilibrium there is no theory for knowing when from it bifurcates some small-amplitude limit cycle 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 differential system in dimension $3$ or higher. In this paper first 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 sufficient conditions for the bifurcation of limit cycles from these families of zero-Hopf equilibria. From one family we can prove that $1$ limit cycle bifurcate, and from the other two families we can prove that $1$, $2$ or $3$ limit cycles bifurcate simultaneously.