A trajectory-driven algorithm for differentiating SRB measures on unstable manifolds

TL;DR

This paper introduces a new trajectory-driven algorithm to compute SRB density gradients on unstable manifolds, enhancing sensitivity analysis in chaotic systems with high-dimensional unstable directions.

Contribution

The paper presents a novel, memory-efficient algorithm leveraging measure preservation and the chain rule, suitable for high-dimensional unstable manifolds in chaotic dynamics.

Findings

Exponential convergence demonstrated numerically

Algorithm integrates with Monte Carlo schemes

Analyzed computational cost and efficiency

Abstract

SRB measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure densities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-products of the regularization of Lebesgue integrals appearing in the original linear response expression. In this paper, we propose a novel trajectory-driven algorithm for computing the SRB density gradient defined for systems with high-dimensional unstable manifolds. We apply the concept of measure preservation together with the chain rule on smooth manifolds. Due to the recursive one-step nature of our derivations, the proposed procedure is memory-efficient and can be naturally integrated with existing Monte Carlo schemes widely used in computational chaotic dynamics. We numerically show the exponential convergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.