Abstract
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.
Similar content being viewed by others
Data Availability Statement
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Albers, P., Mazzucchelli, M.: Periodic bounce orbits of prescribed energy. Int. Math. Res. Not. IMRN. 2011(14), 3289–3314 (2012)
Andresand, J., Górniewicz, L.: Periodic solutions of dissipative systems revisited. Fixed Point Theory Appl. 2006(1), 1–12 (2005)
Benci, V.: Normal modes of a Lagrangian system constrained in a potential well. Ann. Inst. H. Poincaré Anal. Non Linéaire 1(5), 379–400 (1984)
Benci, V., Giannoni, F.: Periodic bounce trajectories with a low number of bounce points. Ann. Inst. H. Poincaré Anal. Non Linéaire 6(1), 73–93 (1989)
Buttazzo, G., Percivale, D.: On the approximation of the elastic bounce problem on Riemannian manifolds. J. Differential Equations 47(2), 227–245 (1983)
Cao, Z.B., Zhang, X.M., Li, D.H., Yin, S., Xie, J.H.: Existence of invariant curves for a Fermi-type impact absorber. Nonlinear Dyn. 99(4), 2647–2656 (2020)
Chernov, N., Markarian, R.: Chaotic Billiards. AMS, Providence (2006)
Evans, L.C.: Partial Differential Equations. AMS, Providence (2010)
Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (2013)
Marò, S.: Chaotic dynamics in an impact problem. Ann. Henri Poincaré 16(7), 1633–1650 (2015)
Marò, S.: Diffusion and chaos in a bouncing ball model. Z. Angew. Math. Phys. 71(78), 1–18 (2020)
Percivale, D.: Uniqueness in the elastic bounce problem. J. Differ. Equa. 56(2), 206–215 (1985)
Rapoport, A., Rom-Kedar, V., Turaev, D.: Approximating multi-dimensional Hamiltonian flows by billiards. Comm. Math. Phys. 272(3), 567–600 (2007)
Rapoport, A., Rom-Kedar, V., Turaev, D.: Billiards: a singular perturbation limit of smooth Hamiltonian flows. Chaos 22(2), 026102 (2012)
Tabachnikov, S.: Geometry and Billiards. AMS, Providence (2005)
Yoshizawa, T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions. Springer, New York (2012)
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11732014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, X., Cao, Z., Li, D. et al. Existence of Periodic Solutions in the Systems of the Billiard Type. Qual. Theory Dyn. Syst. 20, 72 (2021). https://doi.org/10.1007/s12346-021-00514-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s12346-021-00514-5