Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum

Xiaoming Zhang; Zhenbang Cao; Denghui Li; Celso Grebogi; Jianhua Xie

doi:10.1016/j.physd.2022.133210

Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum

Xiaoming Zhang, Zhenbang Cao, Denghui Li^* (Corresponding Author), Celso Grebogi, Jianhua Xie

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

4 Citations (Scopus)

Abstract

We investigate a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation. Based on KAM theory, we prove that there are three regions (corresponding to different energies) occupied by quasi-periodic solutions in phase space when the periodic excitation is small. Moreover, the rotational quasi-periodic motion is maintained when the perturbation gets larger. The existence of subharmonic periodic solutions is obtained by the Aubry–Mather theory and the boundedness of all solutions is followed by the fact that there exist abundant invariant tori near infinity. To study the homoclinic bifurcation of this system, we present a numerical method to compute the discontinuous invariant manifolds accurately, which provides a useful tool for the study of invariant manifolds under the effect of impacts.

Original language	English
Article number	133210
Number of pages	14
Journal	Physica D: Nonlinear Phenomena
Volume	434
Early online date	8 Mar 2022
DOIs	https://doi.org/10.1016/j.physd.2022.133210
Publication status	Published - 1 Jun 2022

Bibliographical note

Acknowledgments
The authors are grateful to the anonymous referees for a careful reading and suggestions that led to an improvement of the paper. This work is supported by the National Natural Science Foundation of China (12172306,11732014).

Keywords

KAM theory
Quasi-periodic solution
Impact system
Computation of discontinuous invariant manifold

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Cite this

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title = "Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum",

abstract = "We investigate a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation. Based on KAM theory, we prove that there are three regions (corresponding to different energies) occupied by quasi-periodic solutions in phase space when the periodic excitation is small. Moreover, the rotational quasi-periodic motion is maintained when the perturbation gets larger. The existence of subharmonic periodic solutions is obtained by the Aubry–Mather theory and the boundedness of all solutions is followed by the fact that there exist abundant invariant tori near infinity. To study the homoclinic bifurcation of this system, we present a numerical method to compute the discontinuous invariant manifolds accurately, which provides a useful tool for the study of invariant manifolds under the effect of impacts.",

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author = "Xiaoming Zhang and Zhenbang Cao and Denghui Li and Celso Grebogi and Jianhua Xie",

note = "Acknowledgments The authors are grateful to the anonymous referees for a careful reading and suggestions that led to an improvement of the paper. This work is supported by the National Natural Science Foundation of China (12172306,11732014). ",

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T1 - Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum

AU - Zhang, Xiaoming

AU - Cao, Zhenbang

AU - Li, Denghui

AU - Grebogi, Celso

AU - Xie, Jianhua

N1 - Acknowledgments The authors are grateful to the anonymous referees for a careful reading and suggestions that led to an improvement of the paper. This work is supported by the National Natural Science Foundation of China (12172306,11732014).

PY - 2022/6/1

Y1 - 2022/6/1

N2 - We investigate a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation. Based on KAM theory, we prove that there are three regions (corresponding to different energies) occupied by quasi-periodic solutions in phase space when the periodic excitation is small. Moreover, the rotational quasi-periodic motion is maintained when the perturbation gets larger. The existence of subharmonic periodic solutions is obtained by the Aubry–Mather theory and the boundedness of all solutions is followed by the fact that there exist abundant invariant tori near infinity. To study the homoclinic bifurcation of this system, we present a numerical method to compute the discontinuous invariant manifolds accurately, which provides a useful tool for the study of invariant manifolds under the effect of impacts.

AB - We investigate a nonlinear inverted pendulum impacting between two rigid walls under external periodic excitation. Based on KAM theory, we prove that there are three regions (corresponding to different energies) occupied by quasi-periodic solutions in phase space when the periodic excitation is small. Moreover, the rotational quasi-periodic motion is maintained when the perturbation gets larger. The existence of subharmonic periodic solutions is obtained by the Aubry–Mather theory and the boundedness of all solutions is followed by the fact that there exist abundant invariant tori near infinity. To study the homoclinic bifurcation of this system, we present a numerical method to compute the discontinuous invariant manifolds accurately, which provides a useful tool for the study of invariant manifolds under the effect of impacts.

KW - KAM theory

KW - Quasi-periodic solution

KW - Impact system

KW - Computation of discontinuous invariant manifold

U2 - 10.1016/j.physd.2022.133210

DO - 10.1016/j.physd.2022.133210

M3 - Article

SN - 0167-2789

VL - 434

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

M1 - 133210

ER -

Quasi-periodic solutions and homoclinic bifurcation in an impact inverted pendulum

Abstract

Bibliographical note

Keywords

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