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Statistical properties of type D dispersing billiards
Existence of invariant curves for degenerate almost periodic reversible mappings
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China |
In this paper we are concerned with the existence of invariant curves for almost periodic reversible mappings with higher order degeneracy of the twist condition. In the proof we use a new variant of the KAM theory, containing an artificial parameter $ q, 0<q<1 $, which makes the steps of the KAM iteration infinitely small in the speed of function $ q^n \varepsilon, $ rather than super exponential function.
References:
[1] |
L. Carbajal, D. del-Castillo-Negrete and J. J. Martinell,
Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps, Chaos, 22 (2012), 013137.
doi: 10.1063/1.3694129. |
[2] |
D. del-Castillo-Negrete, Dynamics and Transport in Rotating Fluids and Transition to Chaos in Area Preserving Nontwist Maps, Ph.D thesis, The University of Texas, Austin, (1994). |
[3] |
D. del-Castillo-Negrete, J. M. Greene and P. J. Morrison,
Area preserving nontwist maps: Periodic orbits and transition to chaos, Physica D, 91 (1996), 1-23.
doi: 10.1016/0167-2789(95)00257-X. |
[4] |
D. del-Castillo-Negrete, J. M. Greene and P. J. Morrison,
Renormalization and transition to chaos in area preserving nontwist maps, Physica D, 100 (1997), 311-329.
doi: 10.1016/S0167-2789(96)00200-X. |
[5] |
A. Delshams and R. de la Llave,
KAM theory and a partial justification of Greene's criterion for nontwist maps, SIAM J. Math. Anal., 31 (2000), 1235-1269.
doi: 10.1137/S003614109834908X. |
[6] |
R. Dieckerhoff and E. Zehnder,
Boundedness of solutions via the twist theorem, Ann. Sci. Norm. Sop. Pisa. Cl. Sci., 14 (1987), 79-95.
|
[7] |
S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Berlin: Springer, 1999.
doi: 10.1007/978-1-4471-0869-6. |
[8] |
A. Fischer,
Structure of Fourier exponents of almost periodic functions and periodicity of almost periodic functions, Mathematica Bohemica, 121 (1996), 249-262.
doi: 10.21136/MB.1996.125993. |
[9] |
A. González-Enríquez, A. Haro and R. de la Llave, Singularity Theory for Non-Twist KAM Tori, Mem. Am. Math. Soc., 227 (2014). |
[10] |
M. R. Herman, Surles courbes invariantes par les difféomorphismes de l'anneau I, Astérisque, (1983), 103–104. |
[11] |
M. R. Herman, Surles courbes invariantes par les difféomorphismes de l'anneau II, Astérisque, 144 (1986). |
[12] |
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. |
[13] |
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. |
[14] |
P. Huang, X. Li and B. Liu,
Invariant curves of smooth quasi-periodic mappings, Discrete and Continuous Dynamical Systems, 38 (2018), 131-154.
doi: 10.3934/dcds.2018006. |
[15] |
P. Huang, X. Li and B Liu, Invariant curves of almost periodic twist mappings, J. Dynam. Differential Equations, (2021).
doi: 10.1007/s10884-021-10033-1. |
[16] |
M. Levi,
Quasiperiodic motions in superquadratic time-periodic potrntials, Commun. Math. Phys., 143 (1986), 43-83.
doi: 10.1007/BF02100285. |
[17] |
M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, Smooth Ergodic Theory and its Applications, (Seattle, WA, 1999), (Proc. Symp. Pure Math. vol 69), (Providence, RI: American Mathematical Society) (2001), 733–746.
doi: 10.1090/pspum/069/1858552. |
[18] |
J. E. Littlewood, Some Problems in Real and Complex Analysis, Heath, Lexington Mass., 1968. |
[19] |
B. Liu,
Invariant curves of quasi-periodic reversible mapping, Nonlinearity, 18 (2005), 685-701.
doi: 10.1088/0951-7715/18/2/012. |
[20] |
B. Liu and J. J. Song,
Invariant curves of reversible mappings with small twist, Acta Math. Sin. (Engl. Ser.), 20 (2004), 15-24.
doi: 10.1007/s10114-004-0316-4. |
[21] |
B. Liu and J. You,
Quasiperiodic solutions of Duffing's equations, Nonlinear Anal., 33 (1998), 645-655.
doi: 10.1016/S0362-546X(98)00662-2. |
[22] |
G. R. Morris,
A case of boundedness in Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14 (1976), 71-93.
doi: 10.1017/S0004972700024862. |
[23] |
J. Moser, On invariant curves of area-perserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math. -Phys., vol II (1962), 1–20. |
[24] |
J. Moser,
Combination tones for Duffing's equation, Commun. Pure Appl. Math., 18 (1965), 167-181.
doi: 10.1002/cpa.3160180116. |
[25] |
J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, (1973). |
[26] |
J. Moser,
A stability theorem for minimal foliations on a torus, Ergodic Theory Dynam. Syst., 8 (1988), 251-281.
doi: 10.1017/S0143385700009457. |
[27] |
D. Piao and X. Zhang, Invariant curves of almost periodic reversible mappings, preprint, 2018, arXiv: 1807.06304v3. |
[28] |
J. Pöschel,
Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351-393.
doi: 10.1007/BF02096763. |
[29] |
J. Pöschel, A lecture on the classical KAM theorem, Smooth Ergodic Theory and Its Applications, Proc. Symp. Pure Math., 69 (2001), 707–732.
doi: 10.1090/pspum/069/1858551. |
[30] |
J. Pöschel,
KAM à la R., Regul. Chaotic Dyn., 16 (2011), 17-23.
doi: 10.1134/S1560354710520060. |
[31] |
H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, (1970), 67–105. |
[32] |
H. Rüssmann,
KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character, Discret. Contin. Dyn. Syst. Ser. S, 3 (2010), 683-718.
doi: 10.3934/dcdss.2010.3.683. |
[33] |
M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, vol. 1211. Springer, Berlin, (1986). 180–195.
doi: 10.1007/BFb0075877. |
[34] |
C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Berlin: Springer, 1997. |
[35] |
C. Simó,
Invariant curves of analytic perturbed nontwist area preserving maps, Regul. Chaotic Dyn., 3 (1998), 180-195.
doi: 10.1070/rd1998v003n03ABEH000088. |
[36] |
J. Xu, J. You and Q. Qiu,
Invariant tori of nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-386.
doi: 10.1007/PL00004344. |
[37] |
D. Zhang and J. Xu,
Invariant curves of analytic reversible mappings under Brjuno-Rüssmann's non-resonant condition, J. Dynam. Differential Equations, 26 (2014), 989-1005.
doi: 10.1007/s10884-014-9366-1. |
[38] |
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. Carbajal, D. del-Castillo-Negrete and J. J. Martinell,
Dynamics and transport in mean-field coupled, many degrees-of-freedom, area-preserving nontwist maps, Chaos, 22 (2012), 013137.
doi: 10.1063/1.3694129. |
[2] |
D. del-Castillo-Negrete, Dynamics and Transport in Rotating Fluids and Transition to Chaos in Area Preserving Nontwist Maps, Ph.D thesis, The University of Texas, Austin, (1994). |
[3] |
D. del-Castillo-Negrete, J. M. Greene and P. J. Morrison,
Area preserving nontwist maps: Periodic orbits and transition to chaos, Physica D, 91 (1996), 1-23.
doi: 10.1016/0167-2789(95)00257-X. |
[4] |
D. del-Castillo-Negrete, J. M. Greene and P. J. Morrison,
Renormalization and transition to chaos in area preserving nontwist maps, Physica D, 100 (1997), 311-329.
doi: 10.1016/S0167-2789(96)00200-X. |
[5] |
A. Delshams and R. de la Llave,
KAM theory and a partial justification of Greene's criterion for nontwist maps, SIAM J. Math. Anal., 31 (2000), 1235-1269.
doi: 10.1137/S003614109834908X. |
[6] |
R. Dieckerhoff and E. Zehnder,
Boundedness of solutions via the twist theorem, Ann. Sci. Norm. Sop. Pisa. Cl. Sci., 14 (1987), 79-95.
|
[7] |
S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Berlin: Springer, 1999.
doi: 10.1007/978-1-4471-0869-6. |
[8] |
A. Fischer,
Structure of Fourier exponents of almost periodic functions and periodicity of almost periodic functions, Mathematica Bohemica, 121 (1996), 249-262.
doi: 10.21136/MB.1996.125993. |
[9] |
A. González-Enríquez, A. Haro and R. de la Llave, Singularity Theory for Non-Twist KAM Tori, Mem. Am. Math. Soc., 227 (2014). |
[10] |
M. R. Herman, Surles courbes invariantes par les difféomorphismes de l'anneau I, Astérisque, (1983), 103–104. |
[11] |
M. R. Herman, Surles courbes invariantes par les difféomorphismes de l'anneau II, Astérisque, 144 (1986). |
[12] |
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. |
[13] |
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. |
[14] |
P. Huang, X. Li and B. Liu,
Invariant curves of smooth quasi-periodic mappings, Discrete and Continuous Dynamical Systems, 38 (2018), 131-154.
doi: 10.3934/dcds.2018006. |
[15] |
P. Huang, X. Li and B Liu, Invariant curves of almost periodic twist mappings, J. Dynam. Differential Equations, (2021).
doi: 10.1007/s10884-021-10033-1. |
[16] |
M. Levi,
Quasiperiodic motions in superquadratic time-periodic potrntials, Commun. Math. Phys., 143 (1986), 43-83.
doi: 10.1007/BF02100285. |
[17] |
M. Levi and J. Moser, A Lagrangian proof of the invariant curve theorem for twist mappings, Smooth Ergodic Theory and its Applications, (Seattle, WA, 1999), (Proc. Symp. Pure Math. vol 69), (Providence, RI: American Mathematical Society) (2001), 733–746.
doi: 10.1090/pspum/069/1858552. |
[18] |
J. E. Littlewood, Some Problems in Real and Complex Analysis, Heath, Lexington Mass., 1968. |
[19] |
B. Liu,
Invariant curves of quasi-periodic reversible mapping, Nonlinearity, 18 (2005), 685-701.
doi: 10.1088/0951-7715/18/2/012. |
[20] |
B. Liu and J. J. Song,
Invariant curves of reversible mappings with small twist, Acta Math. Sin. (Engl. Ser.), 20 (2004), 15-24.
doi: 10.1007/s10114-004-0316-4. |
[21] |
B. Liu and J. You,
Quasiperiodic solutions of Duffing's equations, Nonlinear Anal., 33 (1998), 645-655.
doi: 10.1016/S0362-546X(98)00662-2. |
[22] |
G. R. Morris,
A case of boundedness in Littlewood's problem on oscillatory differential equations, Bull. Austral. Math. Soc., 14 (1976), 71-93.
doi: 10.1017/S0004972700024862. |
[23] |
J. Moser, On invariant curves of area-perserving mappings of an annulus, Nachr. Akad. Wiss. Göttingen Math. -Phys., vol II (1962), 1–20. |
[24] |
J. Moser,
Combination tones for Duffing's equation, Commun. Pure Appl. Math., 18 (1965), 167-181.
doi: 10.1002/cpa.3160180116. |
[25] |
J. Moser, Stable and Random Motions in Dynamical Systems, Princeton University Press, Princeton, (1973). |
[26] |
J. Moser,
A stability theorem for minimal foliations on a torus, Ergodic Theory Dynam. Syst., 8 (1988), 251-281.
doi: 10.1017/S0143385700009457. |
[27] |
D. Piao and X. Zhang, Invariant curves of almost periodic reversible mappings, preprint, 2018, arXiv: 1807.06304v3. |
[28] |
J. Pöschel,
Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351-393.
doi: 10.1007/BF02096763. |
[29] |
J. Pöschel, A lecture on the classical KAM theorem, Smooth Ergodic Theory and Its Applications, Proc. Symp. Pure Math., 69 (2001), 707–732.
doi: 10.1090/pspum/069/1858551. |
[30] |
J. Pöschel,
KAM à la R., Regul. Chaotic Dyn., 16 (2011), 17-23.
doi: 10.1134/S1560354710520060. |
[31] |
H. Rüssmann, Kleine Nenner I: Über invariante Kurven differenzierbarer Abbildungen eines Kreisringes, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II, (1970), 67–105. |
[32] |
H. Rüssmann,
KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character, Discret. Contin. Dyn. Syst. Ser. S, 3 (2010), 683-718.
doi: 10.3934/dcdss.2010.3.683. |
[33] |
M. B. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, vol. 1211. Springer, Berlin, (1986). 180–195.
doi: 10.1007/BFb0075877. |
[34] |
C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics, Berlin: Springer, 1997. |
[35] |
C. Simó,
Invariant curves of analytic perturbed nontwist area preserving maps, Regul. Chaotic Dyn., 3 (1998), 180-195.
doi: 10.1070/rd1998v003n03ABEH000088. |
[36] |
J. Xu, J. You and Q. Qiu,
Invariant tori of nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-386.
doi: 10.1007/PL00004344. |
[37] |
D. Zhang and J. Xu,
Invariant curves of analytic reversible mappings under Brjuno-Rüssmann's non-resonant condition, J. Dynam. Differential Equations, 26 (2014), 989-1005.
doi: 10.1007/s10884-014-9366-1. |
[38] |
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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