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Positive Lyapunov exponent for a class of quasi-periodic cocycles
Existence of periodically invariant tori on resonant surfaces for twist mappings
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
In this paper we will prove the existence of periodically invariant tori of twist mappings on resonant surfaces under the low dimensional intersection property.
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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