论文标题
在某些差分系统中极限周期的零-HOPF分叉
Zero-Hopf Bifurcation of Limit Cycles in Certain Differential Systems
论文作者
论文摘要
本文研究了可能从微分方程自主系统平衡中分叉的极限循环的数量。该系统假定为尺寸$ n $，原点具有零HOPF平衡，仅由订单$ m $的均匀条款组成。用$ h_k（n，m）$表示系统的最大限制周期数，可以使用订单$ k $的平均方法检测到。我们证明$ h_1（n，m）\ leq（m-1）\ cdot m^{n-2} $和$ h_k（n，m）\ leq（km）^{n-1} $用于通用$ n \ geq3 $，$ n \ geq3 $，$ m \ geq2 $和$ k> 1 $。数字上的$ H_K（N，M）$或紧密界限的确切数字是通过计算从平均函数获得的某些多项式系统的混合体积来确定的。基于符号和代数计算，提出了一种通用和算法方法，以得出足够的条件，以使给定的差分系统具有规定的极限周期。拟议方法的有效性由三阶微分方程家族和四维高差异差分系统说明。
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.
