This paper is devoted to the study of infinitesimal limit cycles that can bifurcate from zero-Hopf equilibria of differential systems based on the averaging method. We develop an efficient symbolic program using Maple for computing the averaged functions of any order for continuous differential systems in arbitrary dimension. The program allows us to systematically analyze zero-Hopf bifurcations of polynomial differential systems using symbolic computation methods. We show that for the first-order averaging, $\ell\in\{0,1,\ldots,2^{n-3}\}$ limit cycles can bifurcate from the zero-Hopf equilibrium for the general class of perturbed differential systems and up to the second-order averaging, the maximum number of limit cycles can be determined by computing the mixed volume of a polynomial system obtained from the averaged functions. A number of examples are presented to demonstrate the effectiveness of the proposed algorithmic approach.

翻译：本文致力于基于平均方法研究微分系统零-霍普夫平衡点处可分支的无穷小极限环。我们开发了一个高效的Maple符号计算程序，用于计算任意维连续微分系统的任意阶平均函数。该程序使我们能够利用符号计算方法系统地分析多项式微分系统的零-霍普夫分岔。研究表明，对于一阶平均，一般扰动微分系统类从零-霍普夫平衡点可分支出$\ell\in\{0,1,\ldots,2^{n-3}\}$个极限环；而二阶平均时，最大极限环数可通过计算由平均函数所得多项式系统的混合体积确定。文中通过若干算例验证了所提算法方法的有效性。