Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations

TL;DR

This paper introduces a least squares shadowing method to accurately compute sensitivities of long-term averages in chaotic systems, overcoming failures of traditional adjoint methods caused by ill-conditioned problems.

Contribution

It proposes a linearized least squares shadowing approach that reliably computes derivatives of long-term averages in chaotic limit cycle oscillations.

Findings

The method converges to the true derivative of long-term averages.

It effectively handles both periodic and chaotic oscillations.

The approach improves sensitivity analysis accuracy in chaotic dynamical systems.

Abstract

The adjoint method, among other sensitivity analysis methods, can fail in chaotic dynamical systems. The result from these methods can be too large, often by orders of magnitude, when the result is the derivative of a long time averaged quantity. This failure is known to be caused by ill-conditioned initial value problems. This paper overcomes this failure by replacing the initial value problem with the well-conditioned "least squares shadowing (LSS) problem". The LSS problem is then linearized in our sensitivity analysis algorithm, which computes a derivative that converges to the derivative of the infinitely long time average. We demonstrate our algorithm in several dynamical systems exhibiting both periodic and chaotic oscillations.