N-Dimensional Zero-Hopf Bifurcation of Polynomial Differential Systems via Averaging Theory of Second Order

Using the averaging theory of second order, we study the limit cycles which bifurcate from a zero-Hopf equilibrium point of polynomial vector fields with cubic nonlinearities in $\mathbb {R}^{n}$ . We prove that there are at least 3n− 2 limit cycles bifurcating from such zero-Hopf equilibrium points. Moreover, we provide an example in dimension 6 showing that this number of limit cycles is reached.