An augmented shadowing algorithm for calculating the sensitivity of time-average quantities of chaotic systems

Abstract

We apply the non-intrusive least-squares shadowing (NILSS) method to a newly proposed augmented tangent system in order to calculate the sensitivity of time-average quantities of chaotic dynamical systems to parameter variations. The original tangent system is augmented with an additional equation that acts as a low-pass filter, leaving low frequencies unaffected while filtering out high frequencies. A linear damping term is also added to the original tangent system; the term is activated at high frequencies but vanishes at low frequencies. The method introduces two new parameters, the damping coefficient and the time-scale of the filter. Their values can be estimated from the properties of the dynamical system. We evaluate the performance of the proposed algorithm in the Kuramoto–Sivashinsky equation and the Kolmogorov flow system. The number of non-negative Lyapunov exponents (NNLEs) of the augmented system is generally smaller than that of the original system, and this accelerates the sensitivity calculations. Comparisons with the standard NILSS demonstrate the accuracy of the method at a reduced computational cost. The proposed algorithm is more scalable compared to existing approaches and can be applied to sensitivity analysis as well as optimisation and control of complex, large-scale dynamical systems, including turbulent flows.