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

Abstract

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 periodic 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.