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 $m$. Denote by $H_k(n,m)$ the maximum number of limit cycles of the system that can be detected by using the averaging method of order $k$. We prove that $H_1(n,m)\leq(m-1)\cdot m^{n-2}$ and $H_k(n,m)\leq(km)^{n-1}$ for generic $n\geq3$, $m\geq2$ and $k>1$. The exact numbers of $H_k(n,m)$ 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.

翻译：本文研究从自治微分方程组平衡点分岔出的极限环数量。所考虑的系统维数为$n$，原点处具有零-霍普夫平衡点，且仅由$m$次齐次项组成。记$H_k(n,m)$为通过$k$阶平均方法可检测到的系统极限环最大数量。我们证明：对于一般的$n\geq3$、$m\geq2$及$k>1$，有$H_1(n,m)\leq(m-1)\cdot m^{n-2}$和$H_k(n,m)\leq(km)^{n-1}$。通过计算从平均函数得到的若干多项式系统的混合体积，确定了$H_k(n,m)$的精确数值或紧致上界。基于符号与代数计算，本文提出了一种通用算法化方法，用于推导给定微分系统具有指定数量极限环的充分条件。通过一族三阶微分方程和一个四维超混沌微分系统，验证了所提方法的有效性。