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March  2020, 40(3): 1389-1409. doi: 10.3934/dcds.2020081

Existence of periodically invariant tori on resonant surfaces for twist mappings

Lianpeng Yang and  Xiong Li ,

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

* Corresponding author: Xiong Li

Received  January 2019 Revised  June 2019 Published  December 2019

Fund Project: The second author is supported by NSFC (11971059).

In this paper we will prove the existence of periodically invariant tori of twist mappings on resonant surfaces under the low dimensional intersection property.

Keywords: Periodically invariant tori, twist mappings, resonant surfaces, low dimensional intersection property.
Mathematics Subject Classification: Primary: 37E40; Secondary: 37J40.
Citation: Lianpeng Yang, Xiong Li. Existence of periodically invariant tori on resonant surfaces for twist mappings. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1389-1409. doi: 10.3934/dcds.2020081
References:
[1]

V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, in Collected Works, Vladimir I. Arnold - Collected Works, 1, Springer, Berlin, Heidelberg, 2009, 267–294. doi: 10.1007/978-3-642-01742-1_21.

[2]

Q. Bi and J. Xu, Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings, Qual. Theory Dyn. Syst., 13 (2014), 269-288.  doi: 10.1007/s12346-014-0117-9.

[3]

H. Broer, G. Huitema and M. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-540-49613-7.

[4]

C. Q. Cheng and S. Wang, The surviving of lower dimensional tori from a resonant torus of Hamiltonian systems, J. Differential Equations, 155 (1999), 311-326.  doi: 10.1006/jdeq.1998.3586.

[5]

C. Q. Cheng and Y. S. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 275-292.  doi: 10.1007/BF00053456.

[6]

C. Q. Cheng and Y. S. Sun, Existence of periodically invariant curves in 3-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 293-303.  doi: 10.1007/BF00053457.

[7]

F. Cong, Y. Li and M. Huang, Invariant tori for nearly twist mappings with intersection property, Northeast. Math. J., 12 (1996), 280-298. 

[8]

F. Cong, T. Küpper, Y. Li and J. You, KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems, J. Nonlinear Sci., 10 (2000), 49-68.  doi: 10.1007/s003329910003.

[9]

H. R. Dullin and J. D. Meiss, Resonances and twist in volume-preserving mappings, SIAM J. Appl. Dyn. Syst., 11 (2012), 319-349.  doi: 10.1137/110846865.

[10]

L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 57-76.  doi: 10.1007/BF01232935.

[11]

S. M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations, 15 (1974), 1-69.  doi: 10.1016/0022-0396(74)90086-2.

[12]

P. Huang, X. Li and B. Liu, Invariant curves of smooth quasi-periodic mappings, Discrete Contin. Dyn. Syst., 38 (2018), 131-154.  doi: 10.3934/dcds.2018006.

[13]

P. Huang, X. Li and B. Liu, Quasi-periodic solutions for an asymmetric oscillation, Nonlinearity, 29 (2016), 3006-3030.  doi: 10.1088/0951-7715/29/10/3006.

[14]

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

[15]

P. Huang, X. 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.

[16]

A. N. Kolmogorov, On quasi-periodic motions under small perturbations of the Hamiltonian, Dokl. Akas. Nauk SSSR, 98 (1954), 527-530. 

[17]

Y. Li and Y. Yi, A quasi-periodic Poincaré's theorem, Math. Ann., 326 (2003), 649-690.  doi: 10.1007/s00208-002-0399-0.

[18]

Y. Li and Y. Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, Trans. Amer. Math. Soc., 357 (2005), 1565-1600.  doi: 10.1090/S0002-9947-04-03564-0.

[19]

Y. Li and Y. Yi, On Poincaré-Treshchev tori in Hamiltonian systems, in EQUADIFF 2003, World Sci. Publ., Hackensack, NJ, 2005, 136–151. doi: 10.1142/9789812702067_0013.

[20]

A. G. Medvedev, A. I. Neishtadt and D. V. Treschev, Lagrangian tori near resonances of near-integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.  doi: 10.1088/0951-7715/28/7/2105.

[21]

J. Moser, On invariant curves of area-preserving maps of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20. 

[22]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Dover Publications, Inc., New York, 1957.

[23]

M. Rudnev and S. Wiggins, KAM theory near multiplicity one resonant surfaces in perturbations of a-priori stable Hamiltonian systems, J. Nonlinear Sci., 7 (1997), 177-209.  doi: 10.1007/BF02677977.

[24]

C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-87284-6.

[25]

D. V. Treschev, A mechanism of destruction of resonance tori of Hamiltonian systems, Math. USSR-Sb., 68 (1991), 181-203.  doi: 10.1070/SM1991v068n01ABEH001371.

[26]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynam. Systems, 12 (1992), 621-631.  doi: 10.1017/S0143385700006969.

[27]

J. You, Perturbations of lower dimensional tori for Hamiltonian systems, J. Differential Equations, 152 (1999), 1-29.  doi: 10.1006/jdeq.1998.3515.

[28]

W. Zhu, B. Liu and Z. Liu, The hyperbolic invariant tori of symplectic mappings, Nonlinear Anal., 68 (2008), 109-126.  doi: 10.1016/j.na.2006.10.035.

show all references

References:
[1]

V. I. Arnold, Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian, in Collected Works, Vladimir I. Arnold - Collected Works, 1, Springer, Berlin, Heidelberg, 2009, 267–294. doi: 10.1007/978-3-642-01742-1_21.

[2]

Q. Bi and J. Xu, Persistence of lower dimensional hyperbolic invariant tori for nearly integrable symplectic mappings, Qual. Theory Dyn. Syst., 13 (2014), 269-288.  doi: 10.1007/s12346-014-0117-9.

[3]

H. Broer, G. Huitema and M. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Lecture Notes in Mathematics, 1645, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-540-49613-7.

[4]

C. Q. Cheng and S. Wang, The surviving of lower dimensional tori from a resonant torus of Hamiltonian systems, J. Differential Equations, 155 (1999), 311-326.  doi: 10.1006/jdeq.1998.3586.

[5]

C. Q. Cheng and Y. S. Sun, Existence of invariant tori in three-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 275-292.  doi: 10.1007/BF00053456.

[6]

C. Q. Cheng and Y. S. Sun, Existence of periodically invariant curves in 3-dimensional measure-preserving mappings, Celestial Mech. Dynam. Astronom., 47 (1989/90), 293-303.  doi: 10.1007/BF00053457.

[7]

F. Cong, Y. Li and M. Huang, Invariant tori for nearly twist mappings with intersection property, Northeast. Math. J., 12 (1996), 280-298. 

[8]

F. Cong, T. Küpper, Y. Li and J. You, KAM-type theorem on resonant surfaces for nearly integrable Hamiltonian systems, J. Nonlinear Sci., 10 (2000), 49-68.  doi: 10.1007/s003329910003.

[9]

H. R. Dullin and J. D. Meiss, Resonances and twist in volume-preserving mappings, SIAM J. Appl. Dyn. Syst., 11 (2012), 319-349.  doi: 10.1137/110846865.

[10]

L. H. Eliasson, Biasymptotic solutions of perturbed integrable Hamiltonian systems, Bol. Soc. Brasil. Mat. (N.S.), 25 (1994), 57-76.  doi: 10.1007/BF01232935.

[11]

S. M. Graff, On the conservation of hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations, 15 (1974), 1-69.  doi: 10.1016/0022-0396(74)90086-2.

[12]

P. Huang, X. Li and B. Liu, Invariant curves of smooth quasi-periodic mappings, Discrete Contin. Dyn. Syst., 38 (2018), 131-154.  doi: 10.3934/dcds.2018006.

[13]

P. Huang, X. Li and B. Liu, Quasi-periodic solutions for an asymmetric oscillation, Nonlinearity, 29 (2016), 3006-3030.  doi: 10.1088/0951-7715/29/10/3006.

[14]

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

[15]

P. Huang, X. 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.

[16]

A. N. Kolmogorov, On quasi-periodic motions under small perturbations of the Hamiltonian, Dokl. Akas. Nauk SSSR, 98 (1954), 527-530. 

[17]

Y. Li and Y. Yi, A quasi-periodic Poincaré's theorem, Math. Ann., 326 (2003), 649-690.  doi: 10.1007/s00208-002-0399-0.

[18]

Y. Li and Y. Yi, Persistence of lower dimensional tori of general types in Hamiltonian systems, Trans. Amer. Math. Soc., 357 (2005), 1565-1600.  doi: 10.1090/S0002-9947-04-03564-0.

[19]

Y. Li and Y. Yi, On Poincaré-Treshchev tori in Hamiltonian systems, in EQUADIFF 2003, World Sci. Publ., Hackensack, NJ, 2005, 136–151. doi: 10.1142/9789812702067_0013.

[20]

A. G. Medvedev, A. I. Neishtadt and D. V. Treschev, Lagrangian tori near resonances of near-integrable Hamiltonian systems, Nonlinearity, 28 (2015), 2105-2130.  doi: 10.1088/0951-7715/28/7/2105.

[21]

J. Moser, On invariant curves of area-preserving maps of an annulus, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1962 (1962), 1-20. 

[22]

H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Dover Publications, Inc., New York, 1957.

[23]

M. Rudnev and S. Wiggins, KAM theory near multiplicity one resonant surfaces in perturbations of a-priori stable Hamiltonian systems, J. Nonlinear Sci., 7 (1997), 177-209.  doi: 10.1007/BF02677977.

[24]

C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Classics in Mathematics, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-87284-6.

[25]

D. V. Treschev, A mechanism of destruction of resonance tori of Hamiltonian systems, Math. USSR-Sb., 68 (1991), 181-203.  doi: 10.1070/SM1991v068n01ABEH001371.

[26]

Z. Xia, Existence of invariant tori in volume-preserving diffeomorphisms, Ergodic Theory Dynam. Systems, 12 (1992), 621-631.  doi: 10.1017/S0143385700006969.

[27]

J. You, Perturbations of lower dimensional tori for Hamiltonian systems, J. Differential Equations, 152 (1999), 1-29.  doi: 10.1006/jdeq.1998.3515.

[28]

W. Zhu, B. Liu and Z. Liu, The hyperbolic invariant tori of symplectic mappings, Nonlinear Anal., 68 (2008), 109-126.  doi: 10.1016/j.na.2006.10.035.

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