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SRB measures are limiting stationary distributions describing the statisticalbehavior of chaotic dynamical systems. Directional derivatives of SRB measuredensities conditioned on unstable manifolds are critical in the sensitivityanalysis of hyperbolic chaos. These derivatives, known as the SRB densitygradients, are by-products of the regularization of Lebesgue integralsappearing in the original linear response expression. In this paper, we proposea novel trajectory-driven algorithm for computing the SRB density gradientdefined for systems with high-dimensional unstable manifolds. We apply theconcept of measure preservation together with the chain rule on smoothmanifolds. Due to the recursive one-step nature of our derivations, theproposed procedure is memory-efficient and can be naturally integrated withexisting Monte Carlo schemes widely used in computational chaotic dynamics. Wenumerically show the exponential convergence of our scheme, analyze thecomputational cost, and present its use in the context of Monte Carlointegration.

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