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

TL;DR

An efficient and validated method for approximating the stationary measures of random dynamical systems with smooth additive noise and identifies transitions along a hypersurface in parameter space by analyzing the Lyapunov exponent as a function of the system parameters.

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.

Figures (5)

• Figure 1: Parameter region in the $(\alpha, \sigma)$-plane. Blue: $\lambda < 0$, Red: $\lambda > 0$. Rigorous enclosures for the Lyapunov exponent at the points marked with black crosses can be found in Table \ref{['tab:valNIO']}. The Lyapunov exponent was rigorously enclosed on the points of the grid $\alpha = 3 + \frac{i}{1024}$, for $i = 0, \ldots, 1024$, $\sigma = 1/16+ 15/16 \frac{j}{1024}$ for $j = 0, \ldots, 1024$
• Figure 2: Parameter region in the $(\beta, \sigma)$-plane for $\alpha = 3$. Blue: $\lambda < 0$, Red: $\lambda > 0$. Rigorous enclosures for the Lyapunov exponent at the points marked with black crosses can be found in Table \ref{['tab:valNIC']}. The Lyapunov exponent was enclosed rigorously on the points of the grid $\beta = \frac{51}{64} + \frac{i}{8192}$, for $i = 1, \ldots, 1024$, $\sigma = 1/16+ 15/16 \frac{j}{1024}$ for $j = 0, \ldots, 1024$.
• Figure 3: Numerical experiments plotting $|T^i_{\omega}x_0-T^i_{\omega}(y_0)|$ for initial conditions $x_0 = -0.1, y_0 = 0.9$ for noise realizations with $\sigma = 0.1, 0.2971290840221, 0.4$; the transition from positive to negative Lyapunov exponent was rigorously enclosed in $[0.2971290840221, 0.2971290840222]$.
• Figure 4: Enclosure of the transition value $\sigma$ as a function of $\alpha$; in blue the lower bound, in red the upper bound. This figure gives a graphical representation of the definition at which the hypersurface of zero Lyapunov exponent is known in our rigorous data.
• Figure 5: Upper bounds on $\log(||P^N_k|_{\mathcal{U}_0}||_2)$ for $N=2$ and $N=5$ as a function of $\alpha$ and $\sigma$.

Theorems & Definitions (98)

• Remark 2.1
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Definition 4.1
• Definition 4.2
• Lemma 4.3
• proof
• Definition 4.4
• Definition 5.1