Efficient computation of stationary measures and the Lyapunov Landscape for families random dynamical systems with smooth additive noise

TL;DR

This paper introduces an efficient, validated Fourier-based method for approximating stationary measures of random dynamical systems with smooth additive noise, enabling systematic parameter exploration and analysis of noise-induced phenomena.

Contribution

The paper develops a novel Fourier approximation approach with explicit error bounds for computing stationary measures, facilitating rigorous numerical analysis and exploration of noise-induced transitions.

Findings

Identified transitions from positive to negative Lyapunov exponents in parameter space.

Demonstrated the method's efficiency for systematic exploration of noise effects.

Provided rigorous bounds suitable for computer-assisted proofs.

Abstract

We present an efficient and validated method for approximating the stationary measures of random dynamical systems with smooth additive noise. The approach leverages the strong regularizing properties of the associated transfer operator through a finite-dimensional reduction based on Fourier approximation. Explicit error bounds make the method suitable for use in computer-assisted proofs and rigorous numerical investigations; in particular, its efficiency {\em enables systematic explorations of parameter space}. The method provides access to the stationary measure and supports the analysis of key statistical properties of the system. As an application, we study noise-induced phenomena, focusing on the transition from positive to negative Lyapunov exponent (commonly known as Noise Induced Order) in families of random unimodal maps with Gaussian additive noise. By analyzing the Lyapunov exponent as a function of the system parameters, we identify transitions along a hypersurface in parameter space. The parameters we consider include the standard deviation (intensity) of the Gaussian noise and the shape of the unimodal map.