Detecting random bifurcations via rigorous enclosures of large deviations rate functions

TL;DR

The paper develops a rigorous, computer-assisted framework to detect random bifurcations in stochastic flows by analyzing large deviations of finite-time Lyapunov exponents. By linking FTLE statistics to moment Lyapunov exponents through tilted semigroups, it establishes conditions under which the FTLE distribution undergoes a transition from negative to positive and encodes this via the rate function $\mathcal{I}(r)$. Two nonperturbative case studies are treated: a one-dimensional pitchfork bifurcation with additive noise and a two-dimensional linear toy model, with explicit, validated enclosures for $\mathcal{I}(0)$ and derivatives of $\Lambda(p)$, obtained through spectral methods, homotopy, Newton-Kantorovich arguments, and rigorous interval arithmetic. The results demonstrate how the rate function and its minimum reveal the “random bifurcation” behavior and quantify the associated transition probabilities, including the regime where the asymptotic Lyapunov exponent becomes positive. The work thus provides a substantive bridge between stochastic bifurcation theory and large-deviation analysis, with practical computational tools for high-precision spectral enclosures.

Abstract

The main goal of this work is to provide a description of transitions from uniform to non-uniform snychronization in diffusions based on large deviation estimates for finite time Lyapunov exponents. These can be characterized in terms of moment Lyapunov exponents which are principal eigenvalues of the generator of the tilted (Feynman-Kac) semigroup. Using a computer assisted proof, we demonstrate how to determine these eigenvalues and investigate the rate function which is the Legendre-Fenichel transform of the moment Lyapunov function. We apply our results to two case studies: the pitchfork bifurcation and a two-dimensional toy model, also considering the transition to a positive asymptotic Lyapunov exponent.

Figures (4)

• Figure 1: Rigorous enclosures of $\mathcal{I}_\alpha(0)$ for different values of $\alpha$, and $\sigma=1$ in \ref{['eq:pitchforkIntro']}. Upper-bounds are shown in red, and lower bounds in blue, but they are close enough to be hard to distinguish. The right figure is zoomed in near a local minimum at $\alpha \approx 1.2$.
• Figure 2: Numerical computation of $\lambda(\alpha)$ for different values of $\alpha$, and $\sigma=1$ in \ref{['eq:pitchforkIntro']}.
• Figure 3: Representation of the approximate eigenfunction $\bar{f} = \bar{f}(p,\phi)$ (left) and approximate eigenvalue $\bar{\lambda} = \bar{\lambda}(p)$ (right) used in the proof of Theorem \ref{['th:single_sol_shear']}.
• Figure 4: Rigorous computations of $\mathcal{I}_b(0)$, $\Lambda'_b(0)$ and $\Lambda"_b(0)$ for \ref{['eq:specvarEqusim_new_v2']}, with $\alpha=1$, $\sigma=1$ and several values of $b$. The upper- and lower-bounds are too close to be depicted on the picture: all error bounds are below $9\times 10^{-9}$ for $I_b(0)$, below $2\times 10^{-7}$ for $\Lambda'_b(0)$, and below $6\times 10^{-4}$ for $\Lambda"_b(0)$. Blue square markers are used for $\mathcal{I}_b(0)$ when the corresponding asymptotic Lyapunov exponent (given by $\Lambda'_b(0)$) is positive, see Remark \ref{['rem:I0whenlambdapos']}.

Theorems & Definitions (67)

• Theorem 2.3
• proof : Comments on the proof
• Theorem 2.4
• Corollary 2.6
• Remark 2.7
• Theorem 2.10
• Remark 2.11
• Remark 2.12
• Corollary 2.14
• Remark 2.15