Zero-Hopf Bifurcation of Limit Cycles in Certain Differential Systems

This paper studies the number of limit cycles that may bifurcate from an equilibrium of an autonomous system of differential equations. The system in question is assumed to be of dimension $n$, have a zero-Hopf equilibrium at the origin, and consist only of homogeneous terms of order $m$. Using the first-order averaging method, we prove that at most $(m-1)\cdot m^{n-2}$ limit cycles can bifurcate from the origin of the system for generic $n\geq3$ and $m\geq2$. Using the averaging method of order $k$, we show that the system has no more than $(km)^{n-1}$ limit cycles that can bifurcate from the origin. The exact numbers of limit cycles or tight bounds on the numbers are determined by computing the mixed volumes of some polynomial systems obtained from the averaged functions. Based on symbolic and algebraic computation, a general and algorithmic approach is proposed to derive sufficient conditions for a given differential system to have a prescribed number of limit cycles. The effectiveness of the proposed approach is illustrated by a family of third-order differential equations and by a four-dimensional hyperchaotic differential system.