Multistep Methods for Floquet Multipliers and Subspaces

Abstract

Accurate and efficient computation of Floquet multipliers and subspaces is essential for analyzing limit cycles in dynamical systems and periodic steady states in radio frequency circuit simulation. This problem is typically addressed by solving a periodic linear eigenvalue problem, which is obtained by discretizing the linear time-periodic system using one-step collocation methods. Collocation methods become costly for large-scale problems. Our alternative approach is to use multistep methods. A 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 at the consistency order, while the parasitic periodic eigenvalues converge to zero geometrically and hence become separated from the nonzero Floquet multipliers. We design a memory-efficient algorithm, pTOAR, to solve the large-scale pPEP. Its arithmetic and memory costs are almost independent of the choice of multistep methods when the pPEP arises from an implicit multistep discretization. Numerical results agree with our convergence analysis and demonstrate the efficiency of pTOAR.