Multistep Methods for Floquet Multipliers and Subspaces

TL;DR

This paper introduces a multistep method for computing Floquet multipliers and subspaces in dynamical systems, offering higher accuracy and efficiency, especially for large-scale problems, by addressing the limitations of collocation methods.

Contribution

It develops a multistep approach leading to a periodic polynomial eigenvalue problem and proposes a memory-efficient algorithm pTOAR for large-scale computations.

Findings

Convergence of Floquet multipliers improves with smaller stepsize.

Parasitic eigenvalues converge to zero and do not affect Floquet multipliers.

Numerical results confirm theoretical convergence and efficiency of pTOAR.

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 simulation. This problem is typically addressed by solving a periodic linear eigenvalue problem, which is discretized from the linear time-periodic system using one-step collocation methods. Collocation methods become costly for large-scale cases. Our alternative approach is to use multistep methods. The multistep method leads to a periodic polynomial eigenvalue problem (pPEP), and introduces additional parasitic periodic eigenvalues. We prove that as the stepsize decreases, the computed Floquet multipliers and their associated invariant subspace converge with higher order, while the parasitic periodic eigenvalues converge to zero geometrically and therefore Floquet multipliers are not affected by those parasitic ones. A memory-efficient algorithm pTOAR is designed to solve the large-scale pPEP. Its computational and memory costs are almost independent of the choice of multistep methods. Numerical results coincide with our convergence analysis, and also demonstrate the efficiency of pTOAR.