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January  2018, 38(1): 131-154. doi: 10.3934/dcds.2018006

Invariant curves of smooth quasi-periodic mappings

Peng Huang 1, , Xiong Li 1,, and  Bin Liu 2,

1. 

School of Mathematics Sciences, Beijing Normal University, Beijing, 100875, China
2. 

School of Mathematical Sciences, Peking University, Beijing, 100871, China

* Corresponding author

Received  September 2016 Revised  August 2017 Published  September 2017

Fund Project: The second author is supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities. The third author is supported by the NSFC (11231001).

In this paper we are concerned with the existence of invariant curves of planar mappings which are quasi-periodic in the spatial variable, satisfy the intersection property, $\mathcal{C}^{p}$ smooth with $p>2n+1$, $n$ is the number of frequencies.

Keywords: Quasi-periodic mappings, invariant curves.
Mathematics Subject Classification: Primary:37E40, 37J40.
Citation: Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
References:
[1]

L. Chierchia and D. Qian, Moser's theorem for lower dimensional tori, J. Differential Equations, 206 (2004), 55-93.  doi: 10.1016/j.jde.2004.06.014.

[2]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅰ, Astérisque, (1983), 103-104. 

[3]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅱ, Astérisque 144 (1986), 248pp.

[4]

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.

[5]

M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, in Smooth Ergodic Theory and its Applications, (Seattle, WA, 1999) (Proc. Symp. Pure Math. 69), (Providence, RI: American Mathematical Society), 69 (2001), 733-746. doi: 10.1090/pspum/069/1858552.

[6]

B. Liu, Invariant curves of quasi-periodic reversible mapping, Nonlinearity, 18 (2005), 685-701.  doi: 10.1088/0951-7715/18/2/012.

[7]

J. Moser, On invariant curves of area-perserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math. -Phys., 1962 (1962), 1-20. 

[8]

J. Moser, A rapidly convergent iteration method and nonlinear differential equations Ⅱ, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 499-535. 

[9]

J. Moser, A stability theorem for minimal foliations on a torus, Ergod Theory Dynam. Syst., 8 (1988), 251-281.  doi: 10.1017/S0143385700009457.

[10]

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

[11]

H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical systems theory and applications, 38 (1975), 598-624. 

[12]

H. Rüssmann,On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 677-718.

[13]

C. Siegel and J. Moser, Lectures on celestial mechanics, Springer, Berlin, 1995.

[14]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems Ⅰ, Comm. Pure Appl. Math., 28 (1975), 91-140.  doi: 10.1002/cpa.3160280104.

[15]

V. Zharnitsky, Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem, Nonlinearity, 13 (2000), 1123-1136.  doi: 10.1088/0951-7715/13/4/308.

show all references

References:
[1]

L. Chierchia and D. Qian, Moser's theorem for lower dimensional tori, J. Differential Equations, 206 (2004), 55-93.  doi: 10.1016/j.jde.2004.06.014.

[2]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅰ, Astérisque, (1983), 103-104. 

[3]

M. R. Herman, Sur les courbes invariantes par les difféomorphismes de l'anneau Ⅱ, Astérisque 144 (1986), 248pp.

[4]

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.

[5]

M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, in Smooth Ergodic Theory and its Applications, (Seattle, WA, 1999) (Proc. Symp. Pure Math. 69), (Providence, RI: American Mathematical Society), 69 (2001), 733-746. doi: 10.1090/pspum/069/1858552.

[6]

B. Liu, Invariant curves of quasi-periodic reversible mapping, Nonlinearity, 18 (2005), 685-701.  doi: 10.1088/0951-7715/18/2/012.

[7]

J. Moser, On invariant curves of area-perserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math. -Phys., 1962 (1962), 1-20. 

[8]

J. Moser, A rapidly convergent iteration method and nonlinear differential equations Ⅱ, Ann. Scuola Norm. Sup. Pisa, 20 (1966), 499-535. 

[9]

J. Moser, A stability theorem for minimal foliations on a torus, Ergod Theory Dynam. Syst., 8 (1988), 251-281.  doi: 10.1017/S0143385700009457.

[10]

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

[11]

H. Rüssmann, On optimal estimates for the solutions of linear partial differential equations of first order with constant coefficients on the torus, Dynamical systems theory and applications, 38 (1975), 598-624. 

[12]

H. Rüssmann,On the existence of invariant curves of twist mappings of an annulus, Lecture Notes in Math., Springer, Berlin, 1007 (1983), 677-718.

[13]

C. Siegel and J. Moser, Lectures on celestial mechanics, Springer, Berlin, 1995.

[14]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems Ⅰ, Comm. Pure Appl. Math., 28 (1975), 91-140.  doi: 10.1002/cpa.3160280104.

[15]

V. Zharnitsky, Invariant curve theorem for quasiperiodic twist mappings and stability of motion in the Fermi-Ulam problem, Nonlinearity, 13 (2000), 1123-1136.  doi: 10.1088/0951-7715/13/4/308.

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