[Paper Review] A role of random slow manifolds in detecting stochastic bifurcation

Ziying He, Rui Cai+2 more|arXiv (Cornell University)|May 12, 2018

stochastic dynamics and bifurcation16 references3 citations

TL;DR

This paper establishes that random slow manifolds in multi-scale stochastic dynamical systems preserve stochastic equilibrium states and bifurcation phenomena from the original high-dimensional system. By applying singular perturbation to derive an approximate random slow manifold, the authors reduce the system to a lower-dimensional form that accurately captures stochastic bifurcations, enabling efficient detection of complex dynamics without full system simulation.

ABSTRACT

We consider the relation for the stochastic equilibrium states between the reduced system on a random slow manifold and the original system. This provides a theoretical basis for the reduction about sophisti- cated detailed models by the random slow manifold without significant damage to the overall qualitative properties. Based on this result, we reveal a role of random slow manifolds in detecting stochastic bifurca- tion by an example. The example exhibits a stochastic bifurcation phenomenon and possesses a random slow manifold that carries the stochastic bifurcation information of the system. Specifically, the lower dimensional reduced system on the random slow manifold retains the stochastic bifurcation phenomenon from the original system.

Motivation & Objective

• To establish a theoretical link between stochastic equilibrium states in the original high-dimensional system and those on the random slow manifold.
• To investigate whether stochastic bifurcations in the original system are preserved in the reduced system on the random slow manifold.
• To develop a practical method for detecting stochastic bifurcations via low-dimensional dynamics on the random slow manifold.
• To validate the theoretical findings through a concrete example with numerical simulations.
• To demonstrate that model reduction via random slow manifolds retains key qualitative features, including bifurcation structure, for stochastic systems.

Proposed method

• Utilizes singular perturbation theory to derive an explicit approximate expression for the random slow manifold in a two-time-scale stochastic system.
• Applies the Lyapunov-Perron method to construct the random slow manifold as a graph over the slow variable, ensuring exponential tracking of solutions.
• Transforms the original system via a change of variables to decouple slow and fast dynamics, facilitating analysis.
• Derives a reduced one-dimensional stochastic differential equation on the random slow manifold, retaining the essential dynamics.
• Employs the stochastic implicit Euler scheme for numerical simulation of both original and reduced systems to compare bifurcation behavior.
• Uses the gap condition, exponential dichotomy, and Lipschitz continuity to ensure existence and stability of the random slow manifold.

Experimental results

Research questions

• RQ1 Can the stochastic equilibrium states of the original high-dimensional system be preserved in the reduced system on the random slow manifold?
• RQ2 Does the reduced system on the random slow manifold inherit the stochastic bifurcation structure of the original system?
• RQ3 To what extent can the random slow manifold serve as a reliable low-dimensional surrogate for detecting stochastic bifurcations?
• RQ4 How does the approximation error in the random slow manifold affect the fidelity of the reduced system’s bifurcation behavior?
• RQ5 Can the singular perturbation method reliably generate a reduced system that captures the qualitative dynamics of the original system?

Key findings

• The reduced one-dimensional system on the random slow manifold preserves the same number of stochastic equilibrium states as the original system for each parameter value of $ a $.
• The positions of stable equilibrium states in the reduced system closely match those in the original system, confirming structural fidelity.
• Numerical simulations show that both the original and reduced systems exhibit identical stochastic bifurcation patterns as $ a $ varies, including transitions between one and three stable states.
• For $ a = 0.6 $, both systems have one stable equilibrium; for $ a = -0.006 $, both exhibit two stable equilibria, confirming bifurcation consistency.
• The approximate random slow manifold is derived up to $ o(Cvarepsilon^2) $, with explicit terms involving noise integral $ \int_{-\infty}^{t} \tau e^{2\tau} d B_{\tau} $, ensuring analytical tractability.
• The method enables detection of stochastic bifurcations through analysis of the low-dimensional reduced system, significantly simplifying the study of complex stochastic dynamics.