Abstract
In this paper, we consider a mechanical counterexample which satisfies all conditions of Moser’s twist theorem except the smoothness condition. The model (also called ping-pong model) is the vertical motion of a bouncing ball on a plate which moves in the vertical direction as a \(C^{1+\alpha }\)-smooth periodic function with \(0\le \alpha <1/3\). We construct an unbounded orbit of this simple mechanical model to present a mechanical phenomenon violating Moser’s twist theorem.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Dolgopyat, D.: Bouncing balls in non-linear potentials. Discrete Contin. Dyn. Syst. 22(1), 165–182 (2008)
Everson, R.M.: Chaotic dynamics of a bouncing ball. Phys. D 19, 355–383 (1986)
Herman, M.R.: Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 2. (With a correction to: Sur les courbes invariantes par les difféomorphismes de l’anneau. Vol. 1. Astérisque (1983), 103–104.) Astérisque, 144, 248 (1986)
Kunze, M., Ortega, R.: Complete orbits for twist maps on the plane: extensions and applications. J. Dyn. Differ. Equ. 23(3), 405–423 (2011)
Ma, Z., Xu, J.: A Moser’s non-twist theorem for nearly integrable mappings with self-intersection property. Accepted by Proc. Am. Math. Soc
Marò, S.: Coexistence of bounded and unbounded motions in a bouncing ball model. Nonlinearity 26(5), 1439–1448 (2013)
Marò, S.: A mechanical counterexample to KAM theory with low regularity. Phys. D 283, 10–14 (2014)
Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II 6, 1–20 (1962)
Moser, J.: Stable and Random Motion in Dynamical Systems with Special Emphasis on Celestial Mechanica. Ann of Math Stud., vol. 77. Princeton University Press, Princeton (1973)
Pustyl’nikov, L.D.: The existence of a set of positive measure of oscillating motions in a certain problem of dynamics. Dokl. Akad. Nauk SSSR 202, 287–289 (1972)
Qian, D., Torres, P.J.: Periodic motions of linear impact oscillators via the successor map. SIAM J. Math. Anal. 36(6), 1707–1725 (2005)
Rüssman, H., Kleine Nenner, I.: Über invariante kurven differenzierbarer abbildungen eines kreisringes. Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 67–105 (1970)
Rüssmann, H.: On Twist Hamiltonians. Talk on the Colloque International. Mécanique Cèleste et Systèmes Hamiltoniens, Marseille (1990)
Rüssmann, H.: Invariant tori in non-degenerate nearly integrable Hamiltonian systems. Regul. Chaotic Dyn. 6(2), 119–204 (2001)
Takens, F.: A \(C^{1}\) counterexample to Moser’s twist theorem. Nederl. Akad. Wetensch. Proc. Ser. A \(74\)=Indag. Math. 33, 378–386 (1971)
Xu, J., You, J., Qiu, Q.: Invariant tori for nearly integrable Hamiltonian systems with degeneracy. Math. Z. 226(3), 375–387 (1997)
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.
The work was supported by the National Natural Science Foundation of China (11871146).
Rights and permissions
About this article
Cite this article
Ma, Z., Xu, J. A \(C^{1+\alpha }\) mechanical counterexample to Moser’s twist theorem. Z. Angew. Math. Phys. 72, 186 (2021). https://doi.org/10.1007/s00033-021-01618-3
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-021-01618-3