Periodic and quasiperiodic solutions of a forced discontinuous oscillator. (English) Zbl 07916582
Summary: In this paper we consider a forced oscillator with a discontinuous restoring force. By the Aubry-Mather theory we prove that there exist infinitely many periodic and quasiperiodic solutions. The proof relies on analysing the generating function of the system. The approach is applicable to studying the dynamics of more general forced nonsmooth oscillators of Hamiltonian type.
References:
| [1] | Aubry, S.; Le Daeron, PY, The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states, Phys. D, 8, 381-422, 1983 · Zbl 1237.37059 · doi:10.1016/0167-2789(83)90233-6 |
| [2] | Bangert, V., Mather sets for twist maps and geodesics on tori, Dyn. Rep., 1, 1-54, 1988 · Zbl 0664.53021 |
| [3] | Cao, Z.; Grebogi, C.; Li, D.; Xie, J., The existence of Aubry-Mather sets for the Fermi-Ulam model, Qual. Theory Dyn. Syst., 20, 1-12, 2021 · Zbl 1464.37050 · doi:10.1007/s12346-021-00446-0 |
| [4] | Capietto, A.; Soave, N., Some remarks on Mather’s theorem and Aubry-Mather sets, Commun. Appl. Anal., 15, 283-298, 2011 · Zbl 1237.37045 |
| [5] | Chen, H.; Duan, J., Bounded and unbounded solutions of a discontinuous oscillator at resonance, Int. J. Bifur. Chaos Appl. Sci. Eng., 105, 146-151, 2018 |
| [6] | Enguiça, R.; Ortega, R., Functions with average and bounded motions of a forced discontinuous oscillator, J. Dyn. Differ. Equ., 31, 1185-1198, 2019 · Zbl 1443.34020 · doi:10.1007/s10884-017-9595-1 |
| [7] | Jacquemard, A.; Teixeira, MA, Periodic solutions of a class of non-autonomous second order differential equations with discontinuous right-hand side, Phys. D, 241, 2003-2009, 2012 · doi:10.1016/j.physd.2011.05.011 |
| [8] | Katok, A.; Hasselblatt, B., Introduction to the Modern Theory of Dynamical Systems, 1995, Cambridge: Cambridge University Press, Cambridge · Zbl 0878.58020 · doi:10.1017/CBO9780511809187 |
| [9] | Kunze, M.; Küpper, T.; You, J., On the application of KAM theory to discontinuous dynamical systems, J. Differ. Equ., 139, 1-21, 1997 · Zbl 0884.34049 · doi:10.1006/jdeq.1997.3286 |
| [10] | Liu, B., Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231, 355-373, 1999 · Zbl 0926.34026 · doi:10.1006/jmaa.1998.6219 |
| [11] | Mather, JN, Existence of quasiperiodic orbits for twist homeomorphisms of the annulus, Topology, 21, 457-467, 1982 · Zbl 0506.58032 · doi:10.1016/0040-9383(82)90023-4 |
| [12] | Mather, JN; Forni, G.; Graffi, S., Action minimizing orbits in Hamiltonian systems, Transition to Chaos in Classical and Quantum Mechanics. Lecture Notes in Mathematics, 1994, Berlin: Springer, Berlin |
| [13] | Moser, J., Monotone twist mappings and the calculus of variations, Ergodic Theory Dyn. Syst., 6, 3, 401-413, 1986 · Zbl 0619.49020 · doi:10.1017/S0143385700003588 |
| [14] | Marò, S., Coexistence of bounded and unbounded motions in a bouncing ball model, Nonlinearity, 26, 1439-1448, 2013 · Zbl 1281.70027 · doi:10.1088/0951-7715/26/5/1439 |
| [15] | Marò, S.; Ortega, V., Twist dynamics and Aubry-Mather sets around a periodically perturbed point-vortex, J. Differ. Equ., 269, 3624-3651, 2020 · Zbl 1448.37070 · doi:10.1016/j.jde.2020.03.009 |
| [16] | Novaes, DD; Silva, LVMF, Invariant tori and boundedness of solutions of non-smooth oscillators with Lebesgue integrable forcing term, Z. Angew. Math. Phys., 75, 10, 2024 · Zbl 07812525 · doi:10.1007/s00033-023-02152-0 |
| [17] | Ortega, R., Asymmetric oscillators and twist mappings, J. Lond. Math. Soc., 53, 325-342, 1996 · Zbl 0860.34017 · doi:10.1112/jlms/53.2.325 |
| [18] | Percival, I.C.: Variational principles for invariant tori and cantori. In: Month, M., Herrara, J.C. (eds.) Syrup. on Nonlinear Dynamics and Beam-Beam Interactions. No. 57, American Institute of Physics Conf. Proc., pp. 310-320 (1980) |
| [19] | Pustyl’nikov, LD, Existence of invariant curves for maps close to degenerate maps, and a solution of the Fermi-Ulam problem, Russ. Acad. Sci. Sb. Math., 82, 113-124, 1995 · Zbl 0854.58028 |
| [20] | Zhang, X., Xie, J., Li, D., Cao, Z., Grebogi, C.: Stability analysis of the breathing circle billiard. Chaos Solitons Fractals 155, 111643-1-12 (2022) · Zbl 1498.37100 |
| [21] | Zharnitsky, V., Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem, Nonlinearity, 13, 1123-1136, 2000 · Zbl 1004.37026 · doi:10.1088/0951-7715/13/4/308 |
| [22] | Zharnitsky, V., Invariant tori in Hamiltonian systems with impacts, Commun. Math. Phys., 211, 2, 289-302, 2000 · Zbl 0956.37027 · doi:10.1007/s002200050813 |
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