Multistep Methods for Floquet Multipliers and Subspaces

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.