October  2022, 42(10): 4853-4886. doi: 10.3934/dcds.2022074

Existence of invariant curves for degenerate almost periodic reversible mappings

School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025, China

*Corresponding author: Peng Huang

Received  October 2021 Revised  April 2022 Published  October 2022 Early access  June 2022

Fund Project: The author is supported by the National Natural Science Foundation of China (11901131), Guizhou Provincial Science and Technology Foundation ([2020]1Y006)

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.

Citation: Peng Huang. Existence of invariant curves for degenerate almost periodic reversible mappings. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4853-4886. doi: 10.3934/dcds.2022074
References:
[1]

L. CarbajalD. 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-NegreteJ. 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-NegreteJ. 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. HuangX. 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. HuangX. 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. HuangX. 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. XuJ. 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. CarbajalD. 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-NegreteJ. 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-NegreteJ. 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. HuangX. 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. HuangX. 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. HuangX. 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. XuJ. 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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