Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time-Spectral Resolvent Analysis For Periodic Dynamical Systems

Max Howell, Sicheng He

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.

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.
Paper Structure (43 sections, 3 theorems, 124 equations, 12 figures, 1 table)

This paper contains 43 sections, 3 theorems, 124 equations, 12 figures, 1 table.

Table of Contents

  1. Introduction
  2. Steady-State Resolvent Analysis
  3. Dynamical System Setup
  4. Resolvent Operator
  5. Harmonic Resolvent (HR)
  6. Linear Time-Periodic Systems
  7. Harmonic Balance Method
  8. HR Operator Formulation
  9. Time-Spectral Resolvent (TSR)
  10. Time-Spectral Method
  11. TSR Operator Formulation
  12. Normalization of the Forcing and Response Modes
  13. Mapping Modes to the Time Domain
  14. TSR for Autonomous Systems
  15. Null Space of the Time-Spectral Jacobian
  16. ...and 28 more sections

Key Result

Lemma 1

\newlabellem:fourier_decay0 If $g(t)$ is analytic in a strip $|\operatorname{Im} t| < \rho$, then its Fourier coefficients satisfy $\|\hat{g}_k\| \leq C e^{-\alpha|k|}$ for $\alpha < \rho$Trefethen2000boyd2013chebyshev.

Figures (12)

  • Figure 1: The TSR gains perfectly matches the gains calculated from time accurate integration for all forcing frequencies.
  • Figure 2: The optimal forcing (left) and optimal response (right) from the TSR operator. The optimal response perfectly reconstructs the full quasi-periodic response when compared to the simulated result.
  • Figure 3: Relative error of $\sigma_\text{max}(\mathbf{R}_\text{TS})$ with the ground truth gain computed with $501$ collocation points at $\omega_f = \sqrt{2}\omega_0$. The spectral convergence trend can be seen.
  • Figure 4: Base flow of the van der Pol oscillator. The timeseries of the state variables (Left), LCO phase portrait (Middle), and resolved modes for different value of $n_\text{TS}$ (Right) are shown.
  • Figure 5: Resolvent gain (Left), zoom near resonance (Middle), and minimum singular value of $\mathbf{L}_\text{TS}$ (Right) for the van der Pol oscillator.
  • ...and 7 more figures

Theorems & Definitions (5)

  • Lemma 1: Fourier Coefficient Decay
  • Proposition 2: Spectral Convergence of HR
  • Proof 1
  • Theorem 3: Spectral Convergence of TSR
  • Proof 2