High-order Computation of Floquet Multipliers and Subspaces using Multistep Methods

TL;DR

This work introduces a high-order multistep discretization to compute Floquet multipliers and invariant subspaces of linear periodic time-varying systems, formulating a periodic-PEP that maps to a discrete state-transition matrix. It proves convergence: as the stepsize $\Delta t$ decreases, the true Floquet data converge with order $O(\Delta t^{s})$ while spurious modes decay to zero, and it characterizes the spectral separation that enables robust subspace recovery. To solve large problems efficiently, the authors develop pTOAR, a memory-efficient hybrid of periodic Arnoldi and TOAR-style compression, reducing storage and computational costs relative to collocation methods. Numerical experiments on toy models, coupled Stuart–Landau oscillators, and RF circuits confirm high-order accuracy, favorable scaling, and practical viability for data-driven and industrial-scale periodic analyses. The approach advances reliable Floquet analysis in scenarios where system access is discrete or large-scale, with minimal overhead beyond the first-order backward Euler baseline.

Abstract

Accurate and efficient computation of Floquet multipliers and subspaces is essential for analyzing limit cycle in dynamical systems and periodic steady state in Radio Frequency (RF) simulation. This problem is typically addressed by solving a periodic linear eigenvalue problem, which is discretized from the linear periodic time-varying system using one-step methods. The backward Euler method offers a computationally inexpensive overall workflow but has limited accuracy. In contrast, one-step collocation methods achieve higher accuracy through over-sampling, explicit matrix construction, and condensation, thus become costly for large-scale sparse cases. We apply multistep methods to derive a periodic polynomial eigenvalue problem, which introduces additional spurious eigenvalues. Under mild smoothness assumptions, we prove that as the stepsize decreases, the computed Floquet multipliers and their associated invariant subspace converge with higher order, while the spurious eigenvalues converge to zero. To efficiently solve large-scale problems, we propose pTOAR, a memory-efficient iterative algorithm for computing the dominant Floquet eigenpairs. Numerical experiments demonstrate that multistep methods achieves high order accuracy, while its computational and memory costs are only marginally higher than those of the backward Euler method.

Figures (8)

• Figure 1: Convergence Behavior of Dominant Floquet Eigenpair
• Figure 2: Convergence Behavior of Spurious Eigenvalues
• Figure 3: Evolution of Floquet Multipliers vs. Period of the Parameterized Limit Cycle
• Figure 4: Error of Oscillatory Floquet Multiplier vs. Period of the Parameterized Limit Cycle
• Figure 5: Floquet Multipliers, Subspace Separation, and Dominant Eigenvector Convergence for the Well Conditioned Case

Theorems & Definitions (16)

• Definition 1: Multistep Methods hairer1993Multistep[Chapter 3.5]
• Definition 2: Consistency hairer1993Multistep[Chapter 3, 5.17]
• Definition 3: Stability hairer1993Multistep[Chapter 3, Definition 5.4]
• Theorem 1: Bounds of $\operatorname{sep}(A,B)$ XU1997SeperationLowerBoundEstimationSep1985SunjiGuang
• Theorem 2: Invariant Subspace Perturbation Theorem Stewart1971ISbounds
• Definition 4: Periodic Polynomial Eigenvalue Problem
• Lemma 3
• Lemma 4: Stability and Boundedness of Discrete STM
• proof
• Lemma 5