On the Computation of Floquet Multipliers for Periodic Solution in Piecewise-smooth Dynamical System

Abstract

The floquet multiplier is one of the most important indicators for the stability and bifurcation analysis for periodic solutions in nonlinear dynamical systems. Different from the well-established Floquet theory for the perturbation systems of smooth systems, much less has been understood in its counterpart for non-smooth systems. Here in this paper, we will report an unusual and interesting feature of the Floquet multipliers for piecewise-smooth dynamical systems. When the initial condition of the periodic solution is located at the boundary splitting the solution domain, the multipliers would be calculated falsely in certain circumstances, respectively, by a saltation matrix method or a direct numerical integration for the perturbation system. We elucidate the origin of the fake multipliers through perturbation analysis, and furthermore suggest an effective manner to avoid the miscalculation. This finding would be of fundamental significance to both the real-world applications and theory establishment of the Floquet theory in non-smooth systems