The Existence of Aubry–Mather sets for the Fermi–Ulam Model

Zhenbang  Cao; Celso Grebogi; Denghui Li; Jianhua  Xie

doi:10.1007/s12346-021-00446-0

The Existence of Aubry–Mather sets for the Fermi–Ulam Model

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

^*Corresponding author for this work

Southwest Jiaotong University
Hexi University

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

8 Citations (Scopus)

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Abstract

We consider the Fermi–Ulam model, which can be described as a particle moving freely between two vertical rigid walls; the left one being fixed, whereas the right one moves according to a regular periodic function. The particle is elastically reflected when hitting the walls. We show that the dynamics of the model can be described by an area-preserving monotone twist map. Thus, the Aubry–Mather sets exist for every rotation number in the rotation interval. Consequently, this gives a description of global dynamics behavior, particularly a large class of periodic and quasiperiodic orbits for the model.

Original language	English
Article number	12
Number of pages	12
Journal	Qualitative Theory of Dynamical Systems
Volume	20
Early online date	21 Jan 2021
DOIs	https://doi.org/10.1007/s12346-021-00446-0
Publication status	Published - 21 Jan 2021

Bibliographical note

Acknowledgements
This work is supported by the National Natural Science Foundations of China (11732014). The authors express their gratitude to the reviewer for fruitful comments and suggestions.

Keywords

Fermi-Ulam model
Aubry-Mather set
Twist map

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Cite this

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title = "The Existence of Aubry{\textendash}Mather sets for the Fermi{\textendash}Ulam Model",

abstract = "We consider the Fermi{\textendash}Ulam model, which can be described as a particle moving freely between two vertical rigid walls; the left one being fixed, whereas the right one moves according to a regular periodic function. The particle is elastically reflected when hitting the walls. We show that the dynamics of the model can be described by an area-preserving monotone twist map. Thus, the Aubry{\textendash}Mather sets exist for every rotation number in the rotation interval. Consequently, this gives a description of global dynamics behavior, particularly a large class of periodic and quasiperiodic orbits for the model.",

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

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AU - Cao, Zhenbang

AU - Grebogi, Celso

AU - Li, Denghui

AU - Xie, Jianhua

N1 - Acknowledgements This work is supported by the National Natural Science Foundations of China (11732014). The authors express their gratitude to the reviewer for fruitful comments and suggestions.

PY - 2021/1/21

Y1 - 2021/1/21

N2 - We consider the Fermi–Ulam model, which can be described as a particle moving freely between two vertical rigid walls; the left one being fixed, whereas the right one moves according to a regular periodic function. The particle is elastically reflected when hitting the walls. We show that the dynamics of the model can be described by an area-preserving monotone twist map. Thus, the Aubry–Mather sets exist for every rotation number in the rotation interval. Consequently, this gives a description of global dynamics behavior, particularly a large class of periodic and quasiperiodic orbits for the model.

AB - We consider the Fermi–Ulam model, which can be described as a particle moving freely between two vertical rigid walls; the left one being fixed, whereas the right one moves according to a regular periodic function. The particle is elastically reflected when hitting the walls. We show that the dynamics of the model can be described by an area-preserving monotone twist map. Thus, the Aubry–Mather sets exist for every rotation number in the rotation interval. Consequently, this gives a description of global dynamics behavior, particularly a large class of periodic and quasiperiodic orbits for the model.

KW - Fermi-Ulam model

KW - Aubry-Mather set

KW - Twist map

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DO - 10.1007/s12346-021-00446-0

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SN - 1662-3592

VL - 20

JO - Qualitative Theory of Dynamical Systems

JF - Qualitative Theory of Dynamical Systems

M1 - 12

ER -

The Existence of Aubry–Mather sets for the Fermi–Ulam Model

Abstract

Bibliographical note

Keywords

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