Stochastic Bifurcation of Pathwise Random Almost Periodic and Almost Automorphic Solutions for Random Dynamical Systems

TL;DR

This paper introduces and analyzes the bifurcation of pathwise random almost periodic and almost automorphic solutions in stochastic dynamical systems, extending deterministic concepts to stochastic contexts.

Contribution

It develops the theory of pathwise random almost periodic and automorphic solutions and establishes their bifurcation in one-dimensional stochastic equations with multiplicative noise.

Findings

Existence of random almost periodic and automorphic solutions

Bifurcation phenomena for these solutions in stochastic equations

Extension of deterministic concepts to stochastic systems

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.