Abstract
We study nonconservative quasi-periodic (with \(m\) frequencies) perturbations of two-dimensional Hamiltonian systems with nonmonotonic rotation. It is assumed that the perturbation contains the so-called parametric terms. The behavior of solutions in the vicinity of degenerate resonances is described. Conditions for the existence of resonance \((m+1)\)-dimensional invariant tori for which there are no generating ones in the unperturbed system are found. The class of perturbations for which such tori can exist is indicated. The results are applied to the asymmetric Duffing equation under a parametric quasi-periodic perturbation.
This is a preview of subscription content, log in via an institution to check access.
Notes
Some additional conditions are required, see [25].
References
Morozov, A. D. and Shil’nikov, L. P., On Nonconservative Periodic Systems Close to Two-Dimensional Hamiltonian, J. Appl. Math. Mech., 1983, vol. 47, no. 3, pp. 327–334; see also: Prikl. Mat. Mekh., 1983, vol. 47, no. 3, pp. 385-394.
Morozov, A. D., Resonances, Cycles and Chaos in Quasi-Conservative Systems, Izhevsk: R&C Dynamics, 2005 (Russian).
Morozov, A. D. and Boykova, S. A., On the Investigation of Degenerate Resonances, Regul. Chaotic Dyn., 1999, vol. 4, no. 1, pp. 70–82.
Morozov, A. D., Degenerate Resonances in Hamiltonian Systems with \(3/2\) Degrees of Freedom, Chaos, 2002, vol. 12, no. 3, pp. 539–548.
Morozov, A. D., On Degenerate Resonances in Nearly Hamiltonian Systems, Regul. Chaotic Dyn., 2004, vol. 9, no. 3, pp. 337–350.
Morozov, A. D., On Bifurcations in Degenerate Resonance Zones, Regul. Chaotic Dyn., 2014, vol. 19, no. 4, pp. 474–482.
Morozov, A. D., On Degenerate Resonances and “Vortex Pairs”, Regul. Chaotic Dyn., 2008, vol. 13, no. 1, pp. 27–36.
Soskin, S. M., Luchinsky, D. G., Mannella, R., Neiman, A. B., and McClintock, P. V. E., Zero-Dispersion Nonlinear Resonance, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 1997, vol. 7, no. 4, pp. 923–836.
Soskin, S. M., Nonlinear Resonance for the Oscillator with a Nonmonotonic Dependence of Eigenfrequency on Energy, Phys. Rev. E, 1994, vol. 50, no. 1, R44–R46.
Morozov, A. D. and Morozov, K. E., Quasiperiodic Perturbations of Two-Dimensional Hamiltonian Systems with Nonmonotone Rotation, J. Math. Sci. (N. Y.), 2021, vol. 255, no. 6, pp. 741–752.
Morozov, A. D. and Morozov, K. E., Synchronization of Quasiperiodic Oscillations in Nearly Hamiltonian Systems: The Degenerate Case, Chaos, 2021, vol. 31, no. 8, Paper No. 083109, 10 pp.
Morozov, A. D. and Morozov, K. E., Degenerate Resonances and Synchronization in Nearly Hamiltonian Systems under Quasi-Periodic Perturbations, Regul. Chaotic Dyn., 2022, vol. 27, no. 5, pp. 572–585.
Howard, J. E. and Humpherys, J., Nonmonotonic Twist Maps, Phys. D, 1995, vol. 80, no. 3, pp. 256–276.
Petrisor, E., Reconnection Scenarios and the Threshold of Reconnection in the Dynamics of Non-Twist Maps, Chaos Solitons Fractals, 2002, vol. 14, no. 1, pp. 117–127.
Fuchss, K., Wurm, A., Apte, A., and Morrison, P. J., Breakup of Shearless Meanders and “Outer” Tori in the Standard Nontwist Map, Chaos, 2006, vol. 16, no. 3, 033120, 11 pp.
Wurm, A., Apte, A., Fuchss, K., and Morrison, P. J., Meanders and Reconnection-Collision Sequences in the Standard Nontwist Map, Chaos, 2005, vol. 15, no. 2, 023108, 13 pp.
Apte, A., de la Llave, R., and Petrov, N. P., Regularity of Critical Invariant Circles of the Standard Nontwist Map, Nonlinearity, 2005, vol. 18, no. 3, pp. 1173–1187.
Howard, J. E. and Hohs, S. M., Stochasticity and Reconnection in Hamiltonian Systems, Phys. Rev. A (3), 1984, vol. 29, no. 1, pp. 418–421.
del-Castillo-Negrete, D., Greene, J. M., and Morrison, P. J., Area Preserving Nontwist Maps: Periodic Orbits and Transition to Chaos, Phys. D, 1996, vol. 91, no. 1–2, pp. 1–23.
Simó, C., Invariant Curves of Analytic Perturbed Nontwist Area Preserving Maps, Regul. Chaotic Dyn., 1998, vol. 3, no. 3, pp. 180–195.
Dullin, H. R. and Meiss, J. D., Twist Singularities for Symplectic Maps, Chaos, 2003, vol. 13, no. 1, pp. 1–16.
Simó, C. and Vieiro, A., Planar Radial Weakly Dissipative Diffeomorphisms, Chaos, 2010, vol. 20, no. 4, 043138, 18 pp.
Haro, A. and de la Llave, R., A Parametrization Method for the Computation of Invariant Tori and Their Whiskers in Quasi-Periodic Maps: Rigorous Results, J. Differ. Equ., 2006, vol. 228, no. 2, pp. 530–579.
Morozov, A. D. and Morozov, K. E., Quasiperiodic Perturbations of Two-Dimensional Hamiltonian Systems, Differ. Equ., 2017, vol. 53, no. 12, pp. 1557–1566; see also: Differ. Uravn., 2017, vol. 53, no. 12, pp. 1607-1615.
Morozov, A. D. and Morozov, K. E., Global Dynamics of Systems Close to Hamiltonian Ones under Nonconservative Quasi-Periodic Perturbation, Russian J. Nonlinear Dyn., 2019, vol. 15, no. 2, pp. 187–198.
Bogoliubov, N. N. and Mitropolsky, Yu. A., Asymptotic Methods in the Theory of Non-Linear Oscillations, New York: Gordon & Breach, 1961.
Hale, J. K., Ordinary Differential Equations, 2nd ed., Huntington, N.Y.: Krieger, 1980.
Funding
This paper was carried out within the framework of the Russian Ministry of Science and Education [FSWR-2020-0036]. The authors acknowledge support from the Russian Science Foundation under the grant 24-21-00050 (Sections 2–3). The numerical simulations in Section 4 were supported by the RSF (grant 19-11-00280) (Morozov K. E.).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
PUBLISHER’S NOTE
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
MSC2010
34C15, 34C27, 34C37
Rights and permissions
About this article
Cite this article
Morozov, K.E., Morozov, A.D. Quasi-Periodic Parametric Perturbations of Two-Dimensional Hamiltonian Systems with Nonmonotonic Rotation. Regul. Chaot. Dyn. 29, 65–77 (2024). https://doi.org/10.1134/S1560354724010052
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1560354724010052