• Previous Article
    Sharp $ \frac12 $-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles
  • DCDS Home
  • This Issue
  • Next Article
    Global well-posedness to incompressible non-inertial Qian-Sheng model
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 and Continuous Dynamical Systems, 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.

[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.

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.

[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.

[1]

Peng Huang. Existence of invariant curves for degenerate almost periodic reversible mappings. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022074

[2]

Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703

[3]

Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6425-6462. doi: 10.3934/dcdsb.2021026

[4]

Francesca Alessio, Carlo Carminati, Piero Montecchiari. Heteroclinic motions joining almost periodic solutions for a class of Lagrangian systems. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 569-584. doi: 10.3934/dcds.1999.5.569

[5]

Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745

[6]

Tomás Caraballo, David Cheban. Almost periodic and almost automorphic solutions of linear differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1857-1882. doi: 10.3934/dcds.2013.33.1857

[7]

Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268

[8]

Sorin Micu, Ademir F. Pazoto. Almost periodic solutions for a weakly dissipated hybrid system. Mathematical Control and Related Fields, 2014, 4 (1) : 101-113. doi: 10.3934/mcrf.2014.4.101

[9]

Denis Pennequin. Existence of almost periodic solutions of discrete time equations. Discrete and Continuous Dynamical Systems, 2001, 7 (1) : 51-60. doi: 10.3934/dcds.2001.7.51

[10]

Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9

[11]

Xianhua Huang. Almost periodic and periodic solutions of certain dissipative delay differential equations. Conference Publications, 1998, 1998 (Special) : 301-313. doi: 10.3934/proc.1998.1998.301

[12]

Nguyen Minh Man, Nguyen Van Minh. On the existence of quasi periodic and almost periodic solutions of neutral functional differential equations. Communications on Pure and Applied Analysis, 2004, 3 (2) : 291-300. doi: 10.3934/cpaa.2004.3.291

[13]

Xiaocai Wang. Non-floquet invariant tori in reversible systems. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3439-3457. doi: 10.3934/dcds.2018147

[14]

Yong Li, Zhenxin Liu, Wenhe Wang. Almost periodic solutions and stable solutions for stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5927-5944. doi: 10.3934/dcdsb.2019113

[15]

Peter Giesl, Martin Rasmussen. A note on almost periodic variational equations. Communications on Pure and Applied Analysis, 2011, 10 (3) : 983-994. doi: 10.3934/cpaa.2011.10.983

[16]

Marko Kostić. Almost periodic type functions and densities. Evolution Equations and Control Theory, 2022, 11 (2) : 457-486. doi: 10.3934/eect.2021008

[17]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098

[18]

Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1241-1255. doi: 10.3934/dcdsb.2019218

[19]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345

[20]

Amira M. Boughoufala, Ahmed Y. Abdallah. Attractors for FitzHugh-Nagumo lattice systems with almost periodic nonlinear parts. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1549-1563. doi: 10.3934/dcdsb.2020172

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (276)
  • HTML views (99)
  • Cited by (0)

Other articles
by authors

[Back to Top]