Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems

Bixiang Wang

doi:10.3934/dcds.2015.35.3745

Abstract

In this paper, we introduce concepts of pathwise random almost periodic and almost automorphic solutions for dynamical systems generated by non-autonomous stochastic equations. These solutions are pathwise stochastic analogues of deterministic dynamical systems. The existence and bifurcation of random periodic (random almost periodic, random almost automorphic) solutions have been established for a one-dimensional stochastic equation with multiplicative noise.

Highlights

This paper is concerned with almost periodic and almost automorphic dynamics of random dynamical systems associated with stochastic differential equations driven by time-dependent deterministic forcing
Stochastic pitchfork and transcritical bifurcation of these types of solutions for one-dimensional non-autonomous stochastic equations
The first goal of the present paper is to introduce these concepts for random dynamical systems generated by non-autonomous stochastic equations

Summary

Introduction

This paper is concerned with almost periodic and almost automorphic dynamics of random dynamical systems associated with stochastic differential equations driven by time-dependent deterministic forcing. In addition to existence of pathwise random periodic (almost periodic, almost automorphic) solutions, we will study the stability and bifurcation of these solutions. For the bifurcation of stationary solutions of (1.1) with additive noise, we refer the reader to [14] It seems that the bifurcation problem of (1.1) and (1.2) has not been studied in the literature when β and γ are time-dependent. In addition, β and γ are both T -periodic in time for some T > 0, x−λ and x+λ are T -periodic In this case, we obtain pitchfork bifurcation of pathwise random periodic solutions of (1.1). We obtain stochastic pitchfork bifurcation of pathwise random almost periodic (almost automorphic) solutions of (1.1) in this case. In the last two sections, we prove stochastic pitchfork bifurcation and transcritical bifurcation for equations (1.1) and (1.2), respectively

Preliminaries

Pitchfork bifurcation of stochastic equations

Pitchfork bifurcation of a typical non-autonomous stochastic equation

Pitchfork bifurcation of a general non-autonomous stochastic equation

Transcritical bifurcation of stochastic equations