SPACE-SPLIT ALGORITHM FOR SENSITIVITY ANALYSIS OF DISCRETE CHAOTIC SYSTEMS WITH MULTIDIMENSIONAL UNSTABLE MANIFOLDS

Accurate approximations of the change of a system's output and its statistics with respect to the input are highly desired in computational dynamics. Ruelle's linear response theory provides breakthrough mathematical machinery for computing the linear response of chaotic dy-namical systems. In this paper, we propose an algorithm for sensitivity analysis of discrete chaos with an arbitrary number of positive Lyapunov exponents. We combine the concept of perturbation space-splitting, which regularizes Ruelle's original expression, with measure-based parameterization of the expanding subspace. We use these tools to rigorously derive trajectory-following recursive relations that converge exponentially fast and construct a memory-efficient Monte Carlo scheme for derivatives of the output statistics. Thanks to the regularization and lack of simplifying assumptions on the system's behavior, our method is immune to the common problems of other popular methods such as the exploding tangent solutions and unphysical shadowing directions. We provide a ready-to-use algorithm, analyze its complexity, and demonstrate several numerical examples of sensitivity computation using physically inspired low-dimensional systems.

Automated summary

This paper proposes an algorithm for sensitivity analysis of chaotic systems with an arbitrary number of positive Lyapunov exponents. By combining perturbation space-splitting and measure-based parameterization, trajectory-following recursive relations and a memory-efficient Monte Carlo scheme are derived to compute the derivatives of the output statistics accurately.