Time-Spectral Resolvent Analysis For Periodic Dynamical Systems

TL;DR

This paper introduces a novel time-spectral resolvent analysis method for periodic dynamical systems, enabling efficient and accurate identification of amplified responses without complex Fourier coefficient computations.

Contribution

The work develops a time-spectral resolvent operator using Fourier collocation that operates directly in the time domain, simplifying implementation and improving spectral convergence for periodic systems.

Findings

Accurately predicts maximum energy amplification in test systems

Demonstrates spectral convergence and ease of implementation

Validates effectiveness on multiple dynamical systems

Abstract

Traditional resolvent analysis is a powerful framework for identifying the most amplified input-output structures in fluid flows from a stationary base state. Extending this resolvent analysis to periodic base flows poses computational challenges due to quasi-periodic responses and expensive linearization around a time-varying base flow. This work proposes a time-spectral resolvent operator formulated using the time-spectral method and Fourier collocation that operates directly in the time domain. Rather than mapping between truncated Fourier coefficients as in frequency-domain approaches, the proposed operator maps forcing and response envelopes defined on a discrete temporal grid, enabling direct Jacobian evaluation at collocation points without computing Fourier coefficients of the base flow. The time-spectral resolvent achieves spectral convergence and offers simplified implementation that integrates easily with existing scientific computing tools. The time-spectral resolvent method is validated numerically in three examples including the parametrically forced Mathieu oscillator, the autonomous van der Pol oscillator and the complex Ginzburg-Landau partial differential equation to show that the proposed method accurately predicts the maximum energy amplification and optimal response mode when the system is subject to optimal quasi-periodic forcing. The proposed framework provides a foundation for extending resolvent-based analysis and control to high-dimensional periodic dynamical systems.