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Persistence of invariant tori for almost periodically forced reversible systems

  • * Corresponding author: Shengqing Hu

    * Corresponding author: Shengqing Hu

This work was partially supported by the China Postdoctoral Science Foundation (Grant No. 003056)

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  • In this paper, nearly integrable system under almost periodic perturbations is studied

    $ \left\{\begin{array}{l} \dot{x} = \omega_0+y+f(t, x, y), \\ \dot{y} = g(t, x, y), \end{array}\right. $

    where $ x\in\mathbb{T}^n, \, y\in\mathbb{R}^n $, $ \omega_0\in\mathbb{R}^n $ is the frequency vector, and the perturbations $ f, g $ are real analytic almost periodic functions in $ t $ with the infinite frequency $ \omega = (\cdots, \omega_\lambda, \cdots)_{\lambda\in\mathbb{Z}} $. We also assume that the above system is reversible with respect to the involution $ \mathcal{M}_0:(x, y)\rightarrow (-x, y) $. By KAM iterative method, we prove the existence of invariant tori for the above reversible system. As an application, we discuss the existence of almost periodic solutions and the boundedness of all solutions for a second-order nonlinear differential equation.

    Mathematics Subject Classification: Primary: 37J40; Secondary: 70K43, 70H12, 34C11.


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  • [1] V. I. Arnold, Reversible systems, in Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, 1984, 1161–1174.
    [2] H. W. BroerM. C. CiocciH. Hanß mann and A. Vanderbauwhede, Quasi-periodic stability of normally resonant tori, Phys. D, 238 (2009), 309-318.  doi: 10.1016/j.physd.2008.10.004.
    [3] H. W. BroerJ. Hoo and V. Naudot, Normal linear stability of quasi-periodic tori, J. Differential Equations, 232 (2007), 355-418.  doi: 10.1016/j.jde.2006.08.022.
    [4] H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Lecture Notes in Mathematics, Vol. 1645, Springer-Verlag, Berlin, 1996.
    [5] S. Dineen, Complex Analysis on Infinite-Dimensional Spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. doi: 10.1007/978-1-4471-0869-6.
    [6] J. R. Graef, On the generalized Liénard equation with negative damping, J. Differential Equations, 12 (1972), 34-62.  doi: 10.1016/0022-0396(72)90004-6.
    [7] S. Hu and B. Liu, Degenerate lower dimensional invariant tori in reversible system, Discrete Contin. Dyn. Syst., 38 (2018), 3735-3763.  doi: 10.3934/dcds.2018162.
    [8] P. Huang and X. Li, Persistence of invariant tori in integrable Hamiltonian systems under almost periodic perturbations, J. Nonlinear Sci., 28 (2018), 1865-1900.  doi: 10.1007/s00332-018-9467-9.
    [9] P. Huang, X. Li and B. Liu, Invariant curves of almost periodic twist mappings, preprint, arXiv: 1606.08938, 2016.
    [10] P. HuangX. Li and B. Liu, Almost periodic solutions for an asymmetric oscillation, J. Differential Equations, 263 (2017), 8916-8946.  doi: 10.1016/j.jde.2017.08.063.
    [11] M. Levi, Quasi-periodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143 (1991), 43-83.  doi: 10.1007/BF02100285.
    [12] N. Levinson, On the existence of periodic solutions for second order differential equations with a forcing term, J. Math. Phys. Mass. Inst. Tech., 22 (1943), 41-48.  doi: 10.1002/sapm194322141.
    [13] B. Liu, On lower dimensional invariant tori in reversible systems, J. Differential Equations, 176 (2001), 158-194.  doi: 10.1006/jdeq.2000.3960.
    [14] B. Liu and F. Zanolin, Boundedness of solutions of nonlinear differential equations, J. Differential Equations, 144 (1998), 66-98.  doi: 10.1006/jdeq.1997.3355.
    [15] G. R. Morris, A case of boundedness in Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14 (1976), 71-93.  doi: 10.1017/S0004972700024862.
    [16] J. Moser, Stable and Random Motions in Dynamical Systems. With Special Emphasis on Celestial Mechanics, Annals of Mathematics Studies, Vol. 77, Princeton University Press, Princeton, New Jersey, University of Tokyo Press, Tokyo, 1973.
    [17] J. Moser, Quasi-periodic solutions of nonlinear elliptic partial differential equations, Bol. Soc. Brasil. Mat. (N.S.), 20 (1989), 29-45.  doi: 10.1007/BF02585466.
    [18] D. Piao and X. Zhang, Invariant curves of almost periodic reversible mappings, preprint, arXiv: 1807.06304, 2018.
    [19] J. Pöschel, Small divisors with spatial structure in infinite-dimensional Hamiltonian systems, Comm. Math. Phys., 127 (1990), 351-393.  doi: 10.1007/BF02096763.
    [20] G. E. H. Reuter, A boundedness theorem for non-linear differential equations of the second order, Proc. Cambridge Philos. Soc., 47 (1951), 49-54.  doi: 10.1017/S0305004100026360.
    [21] H. Rüssmann, On the one-dimensional Schrödinger equation with a quasiperiodic potential, in Nonlinear Dynamics (Internat. Conf., New York, 1979), Ann. New York Acad. Sci., Vol. 357, New York, 1980, 90–107. doi: 10.1111/j.1749-6632.1980.tb29679.x.
    [22] M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, Vol. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.
    [23] M. B. Sevryuk, Invariant $m$-dimensional tori of reversible systems with a phase space of dimension greater than $2m$, Trudy Sem. Petrovsk., 14 (1989), 109–124,266–267. doi: 10.1007/BF01094996.
    [24] M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565.  doi: 10.1063/1.166125.
    [25] M. B. Sevryuk, New results in the reversible KAM theory, in Seminar on Dynamical Systems (St. Petersburg, 1991), Progr. Nonlinear Differential Equations Appl., Vol. 12, Birkhäuser, Basel, 1994,184–199. doi: 10.1007/978-3-0348-7515-8_14.
    [26] C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York-Heidelberg, 1971.
    [27] X. WangJ. Xu and D. Zhang, Degenerate lower dimensional tori in reversible systems, J. Math. Anal. Appl., 387 (2012), 776-790.  doi: 10.1016/j.jmaa.2011.09.030.
    [28] X. WangJ. Xu and D. Zhang, On the persistence of degenerate lower-dimensional tori in reversible systems, Ergodic Theory Dynam. Systems, 35 (2015), 2311-2333.  doi: 10.1017/etds.2014.34.
    [29] J. G. You, Invariant tori and Lagrange stability of pendulum-type equations, J. Differential Equations, 85 (1990), 54-65.  doi: 10.1016/0022-0396(90)90088-7.
    [30] X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-262.  doi: 10.1006/jdeq.1997.3356.
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