有界噪声扰动下平面微分系统周期轨的分支

Bifurcation of periodic orbit in planar differential systems with bounded random perturbation

本文研究有界噪声扰动下平面微分系统周期轨的分支问题.首先介绍平面微分系统周期轨的三类经典分支:叉型分支、鞍结分支和跨临界分支,并给出其分支方向以及分支出的周期轨的稳定性.然后讨论有界噪声扰动下平面微分系统周期轨的分支现象,考虑该扰动系统极小正向不变集的个数随参数值变化而变化的规律,证明该系统在一定条件下发生所谓的硬分支现象,并给出分支参数值的阶数估计.最后给出具体的例子并进行数值模拟,形象直观地说明本文结果.

The present paper is dedicated to the bifurcation of periodic orbit for a class of planar differential systems with bounded random perturbation. We first recall three kinds of bifurcations of a nonhyperbolic pe- riodic orbit for planar differential systems with a parameter： Pitchfork bifurcation, saddle-node bifurcation and transcritical bifurcation. The direction and the stability of the periodic orbits bifurcating from the nonhyperbolic periodic orbit are investigated. Then we consider a bounded random perturbation on this system, and study the minimal forward invariant （abbreviated as MFI） sets of the perturbed system. When the parameter changes in a small neighborhood of a bifurcation value, we show that the number of MFI sets changes and the hard bifurcation occurs. Some examples are provided to illustrate our theoretical results.