Fast Differentiation of Hyperbolic Chaos

Abstract

We derive the ‘fast response’ formula for the linear response, the parameter derivatives of long-time-averaged statistics, of hyperbolic deterministic and chaotic systems. The expression is pointwisely defined, so we can compute the linear response in high-dimensions via Monte-Carlo-type algorithms. This has two parts, where the shadowing contribution is computed by the nonintrusive shadowing algorithm. The unstable contribution is expressed by renormalized second-order tangent equations; importantly, it does not contain any distributional derivatives. The algorithm’s cost is solving u, the unstable dimension, and many first-order and second-order tangent equations along a long orbit; the main error is the sampling error of the orbit. We numerically demonstrate the algorithm on a 21-dimensional example, which is difficult for previous methods.