Time-Spectral Resolvent Analysis For Periodic 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.