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July  2020, 40(7): 4497-4518. doi: 10.3934/dcds.2020188

Persistence of invariant tori for almost periodically forced reversible systems

Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Shengqing Hu

Received  September 2019 Revised  January 2020 Published  April 2020

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

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.
Citation: Shengqing Hu. Persistence of invariant tori for almost periodically forced reversible systems. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4497-4518. doi: 10.3934/dcds.2020188
References:
[1]

V. I. Arnold, Reversible systems, in Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, 1984, 1161–1174.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Huang, X. Li and B. Liu, Invariant curves of almost periodic twist mappings, preprint, arXiv: 1606.08938, 2016. Google Scholar

[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.  Google Scholar

[11]

M. Levi, Quasi-periodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143 (1991), 43-83.  doi: 10.1007/BF02100285.  Google Scholar

[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.  Google Scholar

[13]

B. Liu, On lower dimensional invariant tori in reversible systems, J. Differential Equations, 176 (2001), 158-194.  doi: 10.1006/jdeq.2000.3960.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Piao and X. Zhang, Invariant curves of almost periodic reversible mappings, preprint, arXiv: 1807.06304, 2018. Google Scholar

[19]

J. Pöschel, Small divisors with spatial structure in infinite-dimensional Hamiltonian systems, Comm. Math. Phys., 127 (1990), 351-393.  doi: 10.1007/BF02096763.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, Vol. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.  Google Scholar

[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.  Google Scholar

[24]

M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565.  doi: 10.1063/1.166125.  Google Scholar

[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.  Google Scholar

[26]

C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-262.  doi: 10.1006/jdeq.1997.3356.  Google Scholar

show all references

References:
[1]

V. I. Arnold, Reversible systems, in Nonlinear and Turbulent Processes in Physics, Vol. 3 (Kiev, 1983), Harwood Academic Publ., Chur, 1984, 1161–1174.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Huang, X. Li and B. Liu, Invariant curves of almost periodic twist mappings, preprint, arXiv: 1606.08938, 2016. Google Scholar

[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.  Google Scholar

[11]

M. Levi, Quasi-periodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143 (1991), 43-83.  doi: 10.1007/BF02100285.  Google Scholar

[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.  Google Scholar

[13]

B. Liu, On lower dimensional invariant tori in reversible systems, J. Differential Equations, 176 (2001), 158-194.  doi: 10.1006/jdeq.2000.3960.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Piao and X. Zhang, Invariant curves of almost periodic reversible mappings, preprint, arXiv: 1807.06304, 2018. Google Scholar

[19]

J. Pöschel, Small divisors with spatial structure in infinite-dimensional Hamiltonian systems, Comm. Math. Phys., 127 (1990), 351-393.  doi: 10.1007/BF02096763.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, Vol. 1211, Springer-Verlag, Berlin, 1986. doi: 10.1007/BFb0075877.  Google Scholar

[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.  Google Scholar

[24]

M. B. Sevryuk, The iteration-approximation decoupling in the reversible KAM theory, Chaos, 5 (1995), 552-565.  doi: 10.1063/1.166125.  Google Scholar

[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.  Google Scholar

[26]

C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Springer-Verlag, New York-Heidelberg, 1971.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

X. Yuan, Invariant tori of Duffing-type equations, J. Differential Equations, 142 (1998), 231-262.  doi: 10.1006/jdeq.1997.3356.  Google Scholar

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