Ergodic Sensitivity Analysis of One-Dimensional Chaotic Maps

TL;DR

This paper introduces the S3 algorithm, a novel ergodic-averaging method for sensitivity analysis in one-dimensional chaotic maps, demonstrating its computational efficiency and theoretical basis in hyperbolic dynamics.

Contribution

The paper presents the S3 algorithm for sensitivity analysis, providing a rigorous ergodic-averaging approach and illustrating its advantages over traditional finite difference methods.

Findings

S3 algorithm effectively computes sensitivities in chaotic maps.

S3 outperforms naive finite difference methods in computational efficiency.

Provides an intuitive explanation of the density gradient component.

Abstract

Sensitivity analysis in chaotic dynamical systems is a challenging task from a computational point of view. In this work, we present a numerical investigation of a novel approach, known as the space-split sensitivity or S3 algorithm. The S3 algorithm is an ergodic-averaging method to differentiate statistics in ergodic, chaotic systems, rigorously based on the theory of hyperbolic dynamics. We illustrate S3 on one-dimensional chaotic maps, revealing its computational advantage over naive finite difference computations of the same statistical response. In addition, we provide an intuitive explanation of the key components of the S3 algorithm, including the density gradient function.