A TRAJECTORY-DRIVEN ALGORITHM FOR DIFFERENTIATING SRB MEASURES ON UNSTABLE MANIFOLDS

Sinai--Ruelle--Bowen (SRB) measures are limiting stationary distributions describing the statistical behavior of chaotic dynamical systems. Directional derivatives of SRB measure den-sities conditioned on unstable manifolds are critical in the sensitivity analysis of hyperbolic chaos. These derivatives, known as the SRB density gradients, are by-pro ducts 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 con-vergence of our scheme, analyze the computational cost, and present its use in the context of Monte Carlo integration.