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

• Zhenbang Cao ; Celso Grebogi ; Denghui Li ^[1] ; Jianhua Xie
  1. [1] Hexi University (Zhangye, China)
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 1, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00446-0
• Enlaces
  • Texto completo
• Resumen

  • 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.

• Referencias bibliográficas
  • 1. Fermi, E.: On the origin of the cosmic radiation. Phys. Rev. 15, 1169–1174 (1949)
  • 2. Ulam, S.: On some statistical properties of dynamical systems. In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics...
  • 3. Douady, R.: Applications du théorème des tores invariants. Thése de 3éme Cycle, University of Paris VII (1982)
  • 4. Pustyl’nikov, L.D.: On Ulam problem. Theor. Math. Phys. 57, 1035–1038 (1983)
  • 5. Pustyl’nikov, L.D.: Existence of invariant curves for maps close to degenerate maps, and a solution of the Fermi–Ulam problem. Russ. Acad....
  • 6. Laederich, S., Levi, M.: Invariant curves and time-dependent potentials. Ergod. Theor. Dyn. Syst. 11, 365–378 (1991)
  • 7. Kruger, T., Pustyl’nikov, L.D., Troubetzkoy, S.E.: Acceleration of bouncing balls in external fields. Nonlinearity 8, 397–410 (1994)
  • 8. Kunze, M., Ortega, R.: Complete orbits for twist maps on the plane: the case of small twist. Ergod. Theor. Dyn. Syst. 31, 1471–1498 (2011)
  • 9. Zharnitsky, V.: Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi–Ulam problem. Nonlinearity...
  • 10. Zharnitsky, V.: Instability in Fermi–Ulam ping-pong problem. Nonlinearity 11, 1481–1487 (1998)
  • 11. de Simoi, J., Dolgopyat, D.: Dynamics of some piecewise smooth Fermi–Ulam models. Chaos 22, 026124 (2012)
  • 12. Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics. Springer, Berlin (1997)
  • 13. Siegel, C.L., Moser, J.: Lectures on Celestial Mechanics. Springer, Berlin (1971)
  • 14. Aubry, S., Le Daeron, P.Y.: The discrete Frenkel–Kontorova model and its extensions. I. Exact results for the ground-states. Phys. D 8,...
  • 15. Mather, J.N.: Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology 21, 457–467 (1982)
  • 16. Boyland, P.: Dual billiards, twist maps and impact oscillators. Nonlinearity 9, 1411–1438 (1996)
  • 17. Katok, A., Hasselblatt, B.: Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge (1995)
  • 18. Marò, S.: Coexistence of bounded and unbounded motions in a bouncing ball model. Nonlinearity 26, 1439–1448 (2013)
  • 19. Pei, M.: Aubry–Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations. J. Differ. Equ. 113, 106–127 (1994)
  • 20. Shi, G.H.: Aubry–Mather sets for relativistic oscillators with anharmonic potentials. Acta Math. Sin. Engl. Ser. 33, 439–448 (2017)
  • 21. Marò, S.: Relativistic pendulum and invariant curves. Discrete Contin. Dyn. Syst. 35, 1139–1162 (2015)
  • 22. Mather, J.N., Forni, G.: Action minimizing orbits in Hamiltonian systems. Transition to Chaos in Classical and Quantum Mechaniccsv (Montecatini...
  • 23. Bangert, V.: Mather sets for twist maps and geodesics on tori. Dyn. Rep. 1, 1–54 (1988)
  • 24. Katok, A.: Some remarks on Birkhoff and Mather twist map theorems. Ergod. Theor. Dyn. Syst. 11, 185–194 (1982)
  • 25. Capietto, A., Soave, N.: Some remarks on Mather’s theorem and Aubry–Mather sets. Commun. Appl. Anal. 15, 283–298 (2011)