Existence of Periodic Solutions in the Systems of the Billiard Type

Zhang, Xiaoming ^[1] ; Cao, Zhenbang ^[1] ; Li, Denghui ^[2] ; Xie, Jianhua ^[1]
1. [1] Southwest Jiaotong University
  
  Southwest Jiaotong University
  
  China
2. [2] Hexi University
  
  Hexi University
  
  China
Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 3, 2021
Idioma: inglés
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Resumen
- In this paper, a billiard type of system is studied. The system describes the motion of a particle in a bounded domain under dissipation and periodic potential. The collisions between the particle and the boundary are elastic. The approximation scheme of Benci and Giannoni (Ann Inst H Poincaré Anal Non Linéaire 6(1), 73–93, 1989) for elastic bounces in conservative systems is extended to time periodic systems with dissipations, and the existence of periodic bounce solutions is proved. Moreover, it is proved that the results still hold when the domain is unbounded, if the potential is coercive. As applications of the results, the existence of periodic bounce orbits are shown in the Fermi model, bouncing ball model with dissipation, and a class of multi-degrees-of-freedom impact systems.