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

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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 languageEnglish
Article number12
Number of pages12
JournalQualitative Theory of Dynamical Systems
Early online date21 Jan 2021
Publication statusPublished - 21 Jan 2021

Bibliographical note

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.


  • Fermi-Ulam model
  • Aubry-Mather set
  • Twist map


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