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Stability Analysis Using Monodromy Matrix for Impacting Systems
Hiroyuki ASAHARA Takuji KOUSAKA
IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences
Publication Date: 2018/06/01
Online ISSN: 1745-1337
Type of Manuscript: PAPER
Category: Nonlinear Problems
hybrid dynamical system, impacting system, periodic border, stability analysis, bifurcation, monodromy matrix, saltation matrix,
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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.