July  2022, 42(7): 3169-3185. doi: 10.3934/dcds.2022013

A KAM theorem for quasi-periodic non-twist mappings and its application

School of Mathematics, Southeast University, Nanjing 210096, China

*Corresponding author: Junxiang Xu

Received  June 2021 Revised  December 2021 Published  July 2022 Early access  February 2022

Fund Project: The authors are supported by the National Natural Science Foundation of China (11871146)

In this paper we consider quasi-periodic non-twist mappings with self-intersection property, which depend on a small parameter. Without assuming any twist condition, we prove that for many sufficiently small parameters the mapping admits an invariant curve. As application, we use this result to study Lagrange stability of second order systems.

Citation: Zhichao Ma, Junxiang Xu. A KAM theorem for quasi-periodic non-twist mappings and its application. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3169-3185. doi: 10.3934/dcds.2022013
References:
[1]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via twist theorem, Ann. Scula. Norm. Sup. Pisa Cl. Sci., 14 (1987), 79-95. 

[2]

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.

[3]

P. HuangX. Li and B. Liu, Invariant curves of smooth quasi-periodic mappings, Discrete Contin. Dyn. Syst., 38 (2018), 131-154.  doi: 10.3934/dcds.2018006.

[4]

L. JiaoD. Piao and Y. Wang, Boundedness for the general semilinear Duffing equations via the twist theorem, J. Differential Equations, 252 (2012), 91-113.  doi: 10.1016/j.jde.2011.09.019.

[5]

S. Laederich and M. Levi, Invariant curves and time-dependent potentials, Ergodic Theory Dynam. Systems, 11 (1991), 365-378.  doi: 10.1017/S0143385700006192.

[6]

M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143 (1991), 43-83.  doi: 10.1007/BF02100285.

[7]

B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.  doi: 10.1006/jmaa.1998.6219.

[8]

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

[9]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 1962 (1962), 1-20. 

[10] J. Moser, Stable and Random Motion in Dynamical Systems, University of Tokyo Press, Tokyo, 1973. 
[11]

J. Moser, Quasi-periodic solutions of nonlinear elliptic partial differential equations, Bol. Soc. Brasil. Mat., 20 (1989), 29-45.  doi: 10.1007/BF02585466.

[12]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.

[13]

R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. Lond. Math. Soc., 79 (1999), 381-413.  doi: 10.1112/S0024611599012034.

[14]

R. Ortega, Invariant curves of mappings with averaged small twist, Adv. Nonlinear Stud., 1 (2001), 14-39.  doi: 10.1515/ans-2001-0102.

[15]

H. Rüssman, Über invariante kurven differenzierbarer abbildungen eines kreisringes, Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 1970 (1970), 67-105. 

[16]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Geometric Dynamics (Rio de Janeiro, 1981), 677–718, Lecture Notes in Math., 1007, Springer, Berlin, 1983. doi: 10.1007/BFb0061441.

[17]

H. Rüssmann, On Twist Hamiltonians, Talk on the Colloque International, Mécanique Cèleste et Systèmes Hamiltoniens, Marseille, 1990.

[18]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.

[19]

Y. Wang, Boundedness of solutions in asymmetric oscillations via the twist theorem, Acta Math. Sin. (Engl. Ser.), 17 (2001), 313-318.  doi: 10.1007/s101140000043.

[20]

J. Xu and X. Lu, General KAM theorems and their applications to invariant tori with prescribed frequencies, Regul Chaotic Dyn., 21 (2016), 107-125.  doi: 10.1134/S1560354716010068.

[21]

J. XuK. Wang and M. Zhu, On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters, Proc. Amer. Math. Soc., 144 (2016), 4793-4805.  doi: 10.1090/proc/13088.

[22]

J. XuJ. You and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-387.  doi: 10.1007/PL00004344.

[23]

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]

R. Dieckerhoff and E. Zehnder, Boundedness of solutions via twist theorem, Ann. Scula. Norm. Sup. Pisa Cl. Sci., 14 (1987), 79-95. 

[2]

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.

[3]

P. HuangX. Li and B. Liu, Invariant curves of smooth quasi-periodic mappings, Discrete Contin. Dyn. Syst., 38 (2018), 131-154.  doi: 10.3934/dcds.2018006.

[4]

L. JiaoD. Piao and Y. Wang, Boundedness for the general semilinear Duffing equations via the twist theorem, J. Differential Equations, 252 (2012), 91-113.  doi: 10.1016/j.jde.2011.09.019.

[5]

S. Laederich and M. Levi, Invariant curves and time-dependent potentials, Ergodic Theory Dynam. Systems, 11 (1991), 365-378.  doi: 10.1017/S0143385700006192.

[6]

M. Levi, Quasiperiodic motions in superquadratic time-periodic potentials, Comm. Math. Phys., 143 (1991), 43-83.  doi: 10.1007/BF02100285.

[7]

B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.  doi: 10.1006/jmaa.1998.6219.

[8]

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

[9]

J. Moser, On invariant curves of area-preserving mappings of an annulus, Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 1962 (1962), 1-20. 

[10] J. Moser, Stable and Random Motion in Dynamical Systems, University of Tokyo Press, Tokyo, 1973. 
[11]

J. Moser, Quasi-periodic solutions of nonlinear elliptic partial differential equations, Bol. Soc. Brasil. Mat., 20 (1989), 29-45.  doi: 10.1007/BF02585466.

[12]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.

[13]

R. Ortega, Boundedness in a piecewise linear oscillator and a variant of the small twist theorem, Proc. Lond. Math. Soc., 79 (1999), 381-413.  doi: 10.1112/S0024611599012034.

[14]

R. Ortega, Invariant curves of mappings with averaged small twist, Adv. Nonlinear Stud., 1 (2001), 14-39.  doi: 10.1515/ans-2001-0102.

[15]

H. Rüssman, Über invariante kurven differenzierbarer abbildungen eines kreisringes, Nachr. Akad. Wiss. Gött. Math.-Phys. Kl. II, 1970 (1970), 67-105. 

[16]

H. Rüssmann, On the existence of invariant curves of twist mappings of an annulus, Geometric Dynamics (Rio de Janeiro, 1981), 677–718, Lecture Notes in Math., 1007, Springer, Berlin, 1983. doi: 10.1007/BFb0061441.

[17]

H. Rüssmann, On Twist Hamiltonians, Talk on the Colloque International, Mécanique Cèleste et Systèmes Hamiltoniens, Marseille, 1990.

[18]

H. Rüssmann, Invariant tori in non-degenerate nearly integrable Hamiltonian systems, Regul. Chaotic Dyn., 6 (2001), 119-204.  doi: 10.1070/RD2001v006n02ABEH000169.

[19]

Y. Wang, Boundedness of solutions in asymmetric oscillations via the twist theorem, Acta Math. Sin. (Engl. Ser.), 17 (2001), 313-318.  doi: 10.1007/s101140000043.

[20]

J. Xu and X. Lu, General KAM theorems and their applications to invariant tori with prescribed frequencies, Regul Chaotic Dyn., 21 (2016), 107-125.  doi: 10.1134/S1560354716010068.

[21]

J. XuK. Wang and M. Zhu, On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters, Proc. Amer. Math. Soc., 144 (2016), 4793-4805.  doi: 10.1090/proc/13088.

[22]

J. XuJ. You and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z., 226 (1997), 375-387.  doi: 10.1007/PL00004344.

[23]

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.

[1]

Lei Wang, Quan Yuan, Jia Li. Persistence of the hyperbolic lower dimensional non-twist invariant torus in a class of Hamiltonian systems. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1233-1250. doi: 10.3934/cpaa.2016.15.1233

[2]

Isaac A. García, Jaume Giné. Non-algebraic invariant curves for polynomial planar vector fields. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 755-768. doi: 10.3934/dcds.2004.10.755

[3]

Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683

[4]

Stefano Marò. Relativistic pendulum and invariant curves. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 1139-1162. doi: 10.3934/dcds.2015.35.1139

[5]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631

[6]

Tifei Qian, Zhihong Xia. Heteroclinic orbits and chaotic invariant sets for monotone twist maps. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 69-95. doi: 10.3934/dcds.2003.9.69

[7]

Marie-Claude Arnaud. A nondifferentiable essential irrational invariant curve for a $C^1$ symplectic twist map. Journal of Modern Dynamics, 2011, 5 (3) : 583-591. doi: 10.3934/jmd.2011.5.583

[8]

Lianpeng Yang, Xiong Li. Existence of periodically invariant tori on resonant surfaces for twist mappings. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1389-1409. doi: 10.3934/dcds.2020081

[9]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234

[10]

Mark Comerford, Todd Woodard. Orbit portraits in non-autonomous iteration. Discrete and Continuous Dynamical Systems - S, 2019, 12 (8) : 2253-2277. doi: 10.3934/dcdss.2019144

[11]

Leonardo Mora. Homoclinic bifurcations, fat attractors and invariant curves. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1133-1148. doi: 10.3934/dcds.2003.9.1133

[12]

Jordi-Lluís Figueras, Àlex Haro. A note on the fractalization of saddle invariant curves in quasiperiodic systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1095-1107. doi: 10.3934/dcdss.2016043

[13]

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

[14]

Fuzhong Cong, Hongtian Li. Quasi-effective stability for a nearly integrable volume-preserving mapping. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1959-1970. doi: 10.3934/dcdsb.2015.20.1959

[15]

Xiao-Bing Li, Xian-Jun Long, Zhi Lin. Stability of solution mapping for parametric symmetric vector equilibrium problems. Journal of Industrial and Management Optimization, 2015, 11 (2) : 661-671. doi: 10.3934/jimo.2015.11.661

[16]

Michael Khanevsky. Non-autonomous curves on surfaces. Journal of Modern Dynamics, 2021, 17: 305-317. doi: 10.3934/jmd.2021010

[17]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413

[18]

Siniša Slijepčević. Stability of invariant measures. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1345-1363. doi: 10.3934/dcds.2009.24.1345

[19]

Peng Huang. Existence of invariant curves for degenerate almost periodic reversible mappings. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022074

[20]

Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (217)
  • HTML views (153)
  • Cited by (0)

Other articles
by authors

[Back to Top]