Stability Analysis Using Monodromy Matrix for Impacting Systems

Hiroyuki ASAHARA; Takuji KOUSAKA

doi:10.1587/transfun.E101.A.904

Stability Analysis Using Monodromy Matrix for Impacting Systems

Hiroyuki ASAHARA, Takuji KOUSAKA

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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.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E101-A No.6 pp.904-914

Publication Date: 2018/06/01

Publicized

Online ISSN: 1745-1337

DOI: 10.1587/transfun.E101.A.904

Type of Manuscript: PAPER

Category: Nonlinear Problems

Authors

Hiroyuki ASAHARA
Okayama University of Science
Takuji KOUSAKA
Chukyo University

Keyword

hybrid dynamical system, impacting system, periodic border, stability analysis, bifurcation, monodromy matrix, saltation matrix

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Hiroyuki ASAHARA, Takuji KOUSAKA, "Stability Analysis Using Monodromy Matrix for Impacting Systems" in IEICE TRANSACTIONS on Fundamentals, vol. E101-A, no. 6, pp. 904-914, June 2018, doi: 10.1587/transfun.E101.A.904.
Abstract: 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.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E101.A.904/_p

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@ARTICLE{e101-a_6_904,
author={Hiroyuki ASAHARA, Takuji KOUSAKA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Stability Analysis Using Monodromy Matrix for Impacting Systems},
year={2018},
volume={E101-A},
number={6},
pages={904-914},
abstract={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.},
keywords={},
doi={10.1587/transfun.E101.A.904},
ISSN={1745-1337},
month={June},}

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TY - JOUR
TI - Stability Analysis Using Monodromy Matrix for Impacting Systems
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 904
EP - 914
AU - Hiroyuki ASAHARA
AU - Takuji KOUSAKA
PY - 2018
DO - 10.1587/transfun.E101.A.904
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E101-A
IS - 6
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - June 2018
AB - 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.
ER -