In this article, we study the zero-Hopf bifurcation of a quartic system in the three-dimensional space which can be obtained from a scalar the third-order ordinary differential equation.

The goal of this paper is to study the number of limit cycles that can bifurcate from the periodic orbits of a linear center perturbed by nonlinear functions inside the class of all generalized Liénard di¤erential equations allowing discontinuities. In particular our results show that for any n 1 there are di¤erential equations of the form x¨+f(x; x_ )x_ +x+sgn(x_ )g(x) = 0, with f and g polynomials of degree n and 1 respectively, having [n=2] + 1 limit cycles, where [] denotes the integer part function.

In this work, we show that a zero–Hopf bifurcation takes place in the di¤erential system as parameters vary. Using averaging theory, we prove the existence of two periodic orbits bifurcating from the zero–Hopf equilibrium for the generalized a Chen–Wang system