Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit error bounds and guaranteed convergence of the Koopman-Hill projection stability method for linear time-periodic dynamics

Fabia Bayer, Remco I. Leine

TL;DR

The paper addresses the lack of explicit error control for the Koopman-Hill projection method in linear time-periodic (LTP) dynamics by deriving a closed-form truncation-error bound that decays exponentially with truncation order \(N\) under the assumption \(\|\mathbf J_k\| \le a e^{-b|k|}\) with \(b>\ln 2\). It also extends the analysis to a subharmonic formulation, showing a faster decay bound \( (2 e^{-b})^{2N} \) and providing practical guidance on choosing the truncation order to guarantee a desired accuracy in computing the fundamental solution matrix and Floquet multipliers. The authors supply constructive series expressions for the true fundamental matrix and the Koopman-Hill approximation, with detailed derivative-based proofs and convergence arguments, and illustrate the theory on the scalar case, the Mathieu equation, and the Duffing oscillator. Overall, this work provides the first rigorous theoretical justification for the Koopman-Hill projection method, facilitating reliable Floquet stability analysis and offering a framework for guaranteed stability in nonlinear periodic problems via linearization.

Abstract

The Koopman-Hill projection method offers an efficient approach for stability analysis of linear time-periodic systems, and thereby also for the Floquet stability analysis of periodic solutions of nonlinear systems. However, its accuracy has previously been supported only by numerical evidence, lacking rigorous theoretical guarantees. This paper presents the first explicit error bound for the truncation error of the Koopman-Hill projection method, establishing a solid theoretical foundation for its application. The bound applies to linear time-periodic systems whose Fourier coefficients decay exponentially with a sufficient rate, and is derived using constructive series expansions. The bound quantifies the difference between the true and approximated fundamental solution matrices, clarifies conditions for guaranteed convergence, and enables conservative but reliable inference of Floquet multipliers and stability properties. Additionally, the same methodology applied to a subharmonic formulation demonstrates improved convergence rates of the latter. Numerical examples, including the Mathieu equation and the Duffing oscillator, illustrate the practical relevance of the bound and underscore its importance as the first rigorous theoretical justification for the Koopman-Hill projection method.

Explicit error bounds and guaranteed convergence of the Koopman-Hill projection stability method for linear time-periodic dynamics

TL;DR

The paper addresses the lack of explicit error control for the Koopman-Hill projection method in linear time-periodic (LTP) dynamics by deriving a closed-form truncation-error bound that decays exponentially with truncation order under the assumption with . It also extends the analysis to a subharmonic formulation, showing a faster decay bound \( (2 e^{-b})^{2N} \) and providing practical guidance on choosing the truncation order to guarantee a desired accuracy in computing the fundamental solution matrix and Floquet multipliers. The authors supply constructive series expressions for the true fundamental matrix and the Koopman-Hill approximation, with detailed derivative-based proofs and convergence arguments, and illustrate the theory on the scalar case, the Mathieu equation, and the Duffing oscillator. Overall, this work provides the first rigorous theoretical justification for the Koopman-Hill projection method, facilitating reliable Floquet stability analysis and offering a framework for guaranteed stability in nonlinear periodic problems via linearization.

Abstract

The Koopman-Hill projection method offers an efficient approach for stability analysis of linear time-periodic systems, and thereby also for the Floquet stability analysis of periodic solutions of nonlinear systems. However, its accuracy has previously been supported only by numerical evidence, lacking rigorous theoretical guarantees. This paper presents the first explicit error bound for the truncation error of the Koopman-Hill projection method, establishing a solid theoretical foundation for its application. The bound applies to linear time-periodic systems whose Fourier coefficients decay exponentially with a sufficient rate, and is derived using constructive series expansions. The bound quantifies the difference between the true and approximated fundamental solution matrices, clarifies conditions for guaranteed convergence, and enables conservative but reliable inference of Floquet multipliers and stability properties. Additionally, the same methodology applied to a subharmonic formulation demonstrates improved convergence rates of the latter. Numerical examples, including the Mathieu equation and the Duffing oscillator, illustrate the practical relevance of the bound and underscore its importance as the first rigorous theoretical justification for the Koopman-Hill projection method.

Paper Structure

This paper contains 33 sections, 16 theorems, 152 equations, 13 figures, 1 table.

Table of Contents

  1. Introduction
  2. Theoretical background and notation
  3. Floquet Theory
  4. The Koopman-Hill projection method motivated through the Koopman framework
  5. The Koopman-Hill projection method motivated through the Koopman framework
  6. Multi-index notation
  7. Main result: Error bound
  8. Improved convergence rate using subharmonic projection
  9. Subharmonic formulation and series expression
  10. Implementation of the subharmonic formulation
  11. Convergence of the subharmonic formulation
  12. Derivative-based proofs for the series expressions
  13. Bound and derivatives of the scalar factor
  14. Proof of Theorem \ref{['thm:proof:Phi_series:xi']}: Series expression for true fundamental matrix
  15. Proof of Theorem \ref{['thm:proof:Q']}: Series expression for Koopman-Hill approximation Q(t)
  16. ...and 18 more sections

Key Result

Theorem 1

If the Fourier coefficient matrices of $\mathbf J(t)$ in Equation eq:background:ode fulfill Assumption assu:b, then the fundamental solution matrix is given by the absolutely convergent series with the scalar, complex-valued factor that is independent of the specific dynamical system and bounded by

Figures (13)

  • Figure 1: Schematic of the Koopman-Hill projection method. The top row illustrates the evolution of $\mathbf y_0$ under the original LTP dynamics and subsequent application of $\mathbf g$ to the evolved state, while the bottom row shows the evolution using the truncated lifted LTI dynamics after application of $\mathbf g$ to the initial condition. If the truncation error is small, the bottom row approximates the top row.
  • Figure 2: Real part (black), imaginary part (gray), and polynomial upper bound (dashed) of the scalar factor $\xi_{\mathbf p}(t)$ over three periods for three exemplary values of $\mathbf p$. Data was computed using Lemma \ref{['lem:proof:xi_p:deriv']} and numerical quadrature with trapezoidal rule.
  • Figure 3: Visualization of the sets $\mathcal{P}_j^{(2)}$ with $N = 3$ for different values of $j$. Center element located at $(j, 0)$ is indicated by a red dot.
  • Figure 4: Visualization of integer index set $\mathcal{P}_0^{(2)}$ (large dots). Inner rhombus (red), outer square (blue) and outer rhombus (black) are indicated.
  • Figure 5: Construction of the subharmonic index set $\mathcal{P}_{\mathrm{subh}}^{(2)}$ as the union of sets $\mathcal{P}_j^{(2)}$, $j = -N, \dots, N$.
  • ...and 8 more figures

Theorems & Definitions (38)

  • Definition 1: Multi-index
  • Example 2.1
  • Definition 2: Products of Fourier coefficient matrices
  • Example 2.2
  • Example 2.3
  • Theorem 1: Series formulation of $\mathbf \Phi(t)$
  • Theorem 2: Series expression for the Koopman-Hill approximations $\mathbf Q(t)$
  • Corollary 3: Direct Koopman-Hill projection
  • proof
  • Theorem 4: Error bound and convergence of the direct Koopman-Hill projection approximation
  • ...and 28 more