August  2022, 42(8): 3931-3951. doi: 10.3934/dcds.2022037

A Moser theorem for multiscale mappings

1. 

College of Mathematics, Jilin University, Changchun 130012, China

2. 

School of Mathematics and Statistics, and Center for Mathematics, and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China

*Corresponding author: Yong Li

Received  September 2021 Revised  January 2022 Published  August 2022 Early access  April 2022

In this paper, we study the persistence of invariant tori in nearly integrable multiscale twist mappings with intersection property and high degeneracy in the integrable part. Such results are also presented for the mappings with distinct number of angles and actions, which affirms the existence of lower-dimensional invariant tori in such mappings. Hence we establish a Moser's theorem in multiscales.

Citation: Xuefeng Zhao, Yong Li. A Moser theorem for multiscale mappings. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3931-3951. doi: 10.3934/dcds.2022037
References:
[1]

V. I. Arnold, Proof of a theorem of A. N. Kolmogorov's theorem on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40. 

[2]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order amidst chaos. Lecture Notes in Mathematics, 1645. Springer-Verlag, Berlin, 1996.

[3]

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.

[4]

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.

[5]

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

[6]

M. R. Herman, Sur le groupe des difféomorphismes du tore, Ann. Inst. Fourier (Grenoble), 23 (1973), 75-86.  doi: 10.5802/aif.457.

[7]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dolk. Akad. Nauk. SSSR(N.S.), 98 (1954), 527-530. 

[8]

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

[9]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[10]

J. Pöschel, A Lecture on the classical KAM theorem, Proc. Sympos. Pure Math., 69 (2001), 707-732.  doi: 10.1090/pspum/069/1858551.

[11]

W. C. QianY. Li and X. Yang, Multiscale KAM theorem for Hamiltonian systems, J. Differential. Equations, 266 (2019), 70-86.  doi: 10.1016/j.jde.2018.07.039.

[12]

H. Rüssmann, Kleine Nenner I. Uber invarianten Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1970 (1970), 67-105. 

[13]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math, 1007 (1983), 677-718.  doi: 10.1007/BFb0061441.

[14]

Z. J. Shang, A note on the KAM theorem for symplectic mappings, J. Dynam. Differential Equations, 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.

[15]

N. V. Svanidze, Small perturbations of an integrable dynamical system with an integral invariant, Trudy Mat. Inst. Steklov, 147 (1980), 124-146. 

[16]

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

[17]

L. XuY. Li and Y. Yi, Lower-dimensional tori in multi-scale, nearly integrable Hamiltonian systems, Ann. Henrri Poincaré, 18 (2017), 53-83.  doi: 10.1007/s00023-016-0516-3.

[18]

L. XuY. Li and Y. Yi, Poincaré-Treshchev mechanism in multi-scale, nearly integrable Hamiltonian systems, J. Nonlinear Sci., 28 (2018), 337-369.  doi: 10.1007/s00332-017-9410-5.

[19]

L. P. Yang and X. Li, Existence of periodically invariant tori on resonant surfaces for twist mappings, Discrete Contin. Dyn. Syst, 40 (2020), 1389-1409.  doi: 10.3934/dcds.2020081.

show all references

References:
[1]

V. I. Arnold, Proof of a theorem of A. N. Kolmogorov's theorem on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian, Uspehi Mat. Nauk, 18 (1963), 13-40. 

[2]

H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems, Order amidst chaos. Lecture Notes in Mathematics, 1645. Springer-Verlag, Berlin, 1996.

[3]

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.

[4]

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.

[5]

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

[6]

M. R. Herman, Sur le groupe des difféomorphismes du tore, Ann. Inst. Fourier (Grenoble), 23 (1973), 75-86.  doi: 10.5802/aif.457.

[7]

A. N. Kolmogorov, On conservation of conditionally periodic motions for a small change in Hamilton's function, Dolk. Akad. Nauk. SSSR(N.S.), 98 (1954), 527-530. 

[8]

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

[9]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696.  doi: 10.1002/cpa.3160350504.

[10]

J. Pöschel, A Lecture on the classical KAM theorem, Proc. Sympos. Pure Math., 69 (2001), 707-732.  doi: 10.1090/pspum/069/1858551.

[11]

W. C. QianY. Li and X. Yang, Multiscale KAM theorem for Hamiltonian systems, J. Differential. Equations, 266 (2019), 70-86.  doi: 10.1016/j.jde.2018.07.039.

[12]

H. Rüssmann, Kleine Nenner I. Uber invarianten Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, 1970 (1970), 67-105. 

[13]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math, 1007 (1983), 677-718.  doi: 10.1007/BFb0061441.

[14]

Z. J. Shang, A note on the KAM theorem for symplectic mappings, J. Dynam. Differential Equations, 12 (2000), 357-383.  doi: 10.1023/A:1009068425415.

[15]

N. V. Svanidze, Small perturbations of an integrable dynamical system with an integral invariant, Trudy Mat. Inst. Steklov, 147 (1980), 124-146. 

[16]

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

[17]

L. XuY. Li and Y. Yi, Lower-dimensional tori in multi-scale, nearly integrable Hamiltonian systems, Ann. Henrri Poincaré, 18 (2017), 53-83.  doi: 10.1007/s00023-016-0516-3.

[18]

L. XuY. Li and Y. Yi, Poincaré-Treshchev mechanism in multi-scale, nearly integrable Hamiltonian systems, J. Nonlinear Sci., 28 (2018), 337-369.  doi: 10.1007/s00332-017-9410-5.

[19]

L. P. Yang and X. Li, Existence of periodically invariant tori on resonant surfaces for twist mappings, Discrete Contin. Dyn. Syst, 40 (2020), 1389-1409.  doi: 10.3934/dcds.2020081.

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