Random attractor and SRB measure for stochastic Hopf bifurcation under discretization

2025/09/16 by Chen, Chuchu, Hong, Jialin, Wang, Yibo
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.12865

Abstract

Chaotic phases in stochastic differential equations are characterized by two essential long-time dynamical features: a random attractor capturing asymptotic geometry and a Sinai-Ruelle-Bowen (SRB) measure describing statistical information. This paper investigates whether the stochastic Hopf bifurcation under discretization could inherit both features. We establish that the stochastic Hopf bifurcation under discretization induces a discrete random dynamical system. Further, we prove that this discrete system possesses a random attractor, and then derive the existence of an SRB measure by demonstrating a strictly positive numerical Lyapunov exponent. Numerical experiments visualize the retained random attractor and SRB measure for the discrete random dynamical system, revealing structures consistent with the theoretical chaotic phase.

Citations

• A new class of splitting methods that preserve ergodicity and exponential integrability for stochastic Langevin equation
• Lyapunov exponents and shear-induced chaos for a Hopf bifurcation with additive noise
• Noise and dissipation in rigid body motion
• Synchronization by noise
• Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise
• Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations

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