American Institute of Mathematical Sciences

Advanced
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 & 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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[1]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[2]

Àlex Haro, Rafael de la Llave. A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1261-1300. doi: 10.3934/dcdsb.2006.6.1261
[3]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[4]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75
[5]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[6]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104
[7]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
[8]

Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169
[9]

Xavier Blanc, Claude Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks & Heterogeneous Media, 2010, 5 (1) : 1-29. doi: 10.3934/nhm.2010.5.1
[10]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
[11]

Jinhao Liang. Positive Lyapunov exponent for a class of quasi-periodic cocycles. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1361-1387. doi: 10.3934/dcds.2020080
[12]

Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209
[13]

Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019
[14]

Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[15]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[16]

Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125
[17]

Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839
[18]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241
[19]

Xuanji Hou, Jiangong You. Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 441-454. doi: 10.3934/dcds.2009.24.441
[20]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (139)
  • HTML views (98)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences