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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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[16]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

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

[27]

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

[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.  Google Scholar

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