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Invariant curves of smooth quasi-periodic mappings
1. | School of Mathematics Sciences, Beijing Normal University, Beijing, 100875, China |
2. | School of Mathematical Sciences, Peking University, Beijing, 100871, China |
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.
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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