Stability Analysis Using Monodromy Matrix for Impacting Systems

Hiroyuki ASAHARA  Takuji KOUSAKA  

Publication
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E101-A   No.6   pp.904-914
Publication Date: 2018/06/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E101.A.904
Type of Manuscript: PAPER
Category: Nonlinear Problems
Keyword: 
hybrid dynamical system,  impacting system,  periodic border,  stability analysis,  bifurcation,  monodromy matrix,  saltation matrix,  

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Summary: 
In this research, we propose an effective stability analysis method to impacting systems with periodically moving borders (periodic borders). First, we describe an n-dimensional impacting system with periodic borders. Subsequently, we present an algorithm based on a stability analysis method using the monodromy matrix for calculating stability of the waveform. This approach requires the state-transition matrix be related to the impact phenomenon, which is known as the saltation matrix. In an earlier study, the expression for the saltation matrix was derived assuming a static border (fixed border). In this research, we derive an expression for the saltation matrix for a periodic border. We confirm the performance of the proposed method, which is also applicable to systems with fixed borders, by applying it to an impacting system with a periodic border. Using this approach, we analyze the bifurcation of an impacting system with a periodic border by computing the evolution of the stable and unstable periodic waveform. We demonstrate a discontinuous change of the periodic points, which occurs when a periodic point collides with a border, in the one-parameter bifurcation diagram.