Mathematical Methods in the Applied Sciences

RESEARCH ARTICLE

Aubry–Mather Sets of the Breathing Circle Billiard

Xiaoming Zhang

[email protected]

orcid.org/0000-0002-1332-7665

School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu, Sichuan, China

State Key Lab of Mechanics and Control for Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China

Contribution: Conceptualization, Methodology, Validation, Writing - original draft

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Denghui Li,

Denghui Li

[email protected]

School of Mathematics and Statistics, Suzhou University of Technology, Changshu, Jiangsu, China

Contribution: Methodology, Supervision, Writing - review & editing, Writing - review & editing, Supervision, Investigation

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Xianbin Liu,

Corresponding Author

Xianbin Liu

[email protected]

State Key Lab of Mechanics and Control for Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China

Correspondence:

Xianbin Liu ([email protected])

Contribution: Writing - review & editing, Supervision, Investigation

Search for more papers by this author

Xiaoming Zhang,

Xiaoming Zhang

[email protected]

orcid.org/0000-0002-1332-7665

School of Mechanics and Aerospace Engineering, Southwest Jiaotong University, Chengdu, Sichuan, China

State Key Lab of Mechanics and Control for Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China

Contribution: Conceptualization, Methodology, Validation, Writing - original draft

Search for more papers by this author

Denghui Li,

Denghui Li

[email protected]

School of Mathematics and Statistics, Suzhou University of Technology, Changshu, Jiangsu, China

Contribution: Methodology, Supervision, Writing - review & editing, Writing - review & editing, Supervision, Investigation

Search for more papers by this author

Xianbin Liu,

Corresponding Author

Xianbin Liu

[email protected]

State Key Lab of Mechanics and Control for Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China

Correspondence:

Xianbin Liu ([email protected])

Contribution: Writing - review & editing, Supervision, Investigation

Search for more papers by this author

First published: 19 May 2025

https://doi.org/10.1002/mma.10996

Funding: This work is supported by the National Natural Science Foundation of China (12302015 and 12172167) and Jiangsu Funding Program for Excellent Postdoctoral Talent.

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ABSTRACT

Using variational techniques, we study the dynamics of the breathing circle billiard. We prove the existence of Aubry–Mather sets, which contain infinite subharmonic and generalized quasi-periodic solutions, in arbitrarily high energy regions. This is under the assumption that the boundary motion is a ${C}^1$ periodic function. Additionally, our method provides a feasible approach to small twist problems and can be applied to some classical impact systems.

Conflicts of Interest

The authors declare no conflicts of interest.

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Volume48, Issue12

August 2025

Pages 11823-11835

Aubry–Mather Sets of the Breathing Circle Billiard

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Aubry–Mather Sets of the Breathing Circle Billiard

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