# American Institute of Mathematical Sciences

July  2021, 26(7): 3905-3919. doi: 10.3934/dcdsb.2020268

## On the reducibility of a class of almost periodic Hamiltonian systems

 a. School of Mathematics and Statistics, Xuzhou Institute of Technology, Xuzhou, 221111, China b. College of Mathematics, Southeast University, Nanjing, 210096, China

Received  October 2019 Revised  July 2020 Published  July 2021 Early access  September 2020

Fund Project: The authors are supported by NSFC grant 11526177 and the Natural Science Foundations for Colleges and Universities in Jiangsu Province grant 18KJB110029

In this paper we consider the following linear almost periodic hamiltonian system
 $\dot{x} = (A+\varepsilon Q(t, \varepsilon))x, \; x\in R^{2},$
where
 $A$
is a constant matrix with different eigenvalues, and
 $Q(t, \varepsilon)$
is analytic almost periodic with respect to
 $t$
and analytic with respect to
 $\varepsilon$
. Without any non-degeneracy condition, we prove that the linear hamiltonian system is reducible for most of sufficiently small parameter
 $\varepsilon$
by an almost periodic symplectic mapping.
Citation: Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268
##### References:
 [1] N. N. Bogoljubov, J. A. Mitropoliski and A. M. Samoilenko, Methods of accelerated convergence in nonlinear mechanics, Springer-Verlag, New York, 1976. [2] L. H. Eliasson, Floquent solutions for the one-dimensional quasi-periodic Schrödinger equation, Commun. Math. Phys., 146 (1992), 447–482. doi: 10.1007/BF02097013. [3] H.-L. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dyn. Differ. Equ., 20 (2008), 831–866. doi: 10.1007/s10884-008-9113-6. [4] R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasiperodic linear differential systems, J. Dyn. Differ. Equ., 41 (1981), 262–288. doi: 10.1016/0022-0396(81)90062-0. [5] A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differ. Equations, 98 (1992), 111–124. doi: 10.1016/0022-0396(92)90107-X. [6] A. Jorba and C. Simó, On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704–1737. doi: 10.1137/S0036141094276913. [7] J. Li and C. Zhu, On the reducibility of a class of finitely differentiable quasi-periodic linear systems, J. Math. Anal. Appl., 413 (2014), 69–83. doi: 10.1016/j.jmaa.2013.10.077. [8] J. Li, C. Zhu and S. Chen, On the reducibility of a class of quasi-periodic Hamiltonian systems with small perturbation parameter near the equilibrium, Qual. Theory Dyn. Syst., 16 (2017), 127–147. doi: 10.1007/s12346-015-0164-x. [9] J. Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351–393. doi: 10.1007/BF02096763. [10] H. Rüssmann, Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities, Ergodic Theor. Dyn. Syst., 24 (2004), 1787–1832. doi: 10.1017/S0143385703000774. [11] X. Wang and J. Xu, On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point, Nonlinear Anal., 69 (2008), 2318–2329. doi: 10.1016/j.na.2007.08.016. [12] H. Whitney, Analytical extensions of differentiable functions defined in closed sets, Trans. A. M. S., 36 (1934), 63–89. doi: 10.1090/S0002-9947-1934-1501735-3. [13] J. Xu and J. You, On reducibility of linear differential equations with almost-periodic coefficients, Chinese Ann. Math. A(in Chinese), 17 (1996), 607–616. [14] J. Xu, On the reducibility of a class of linear differential equations with quasiperiodic coefficients, Mathematika, 46 (1999), 443–451. doi: 10.1112/S0025579300007907. [15] J. Xu and X. Lu, On the reducibility of two-dimensional linear quasi-periodic systems with small parameter, Ergodic Theor. Dyn. Syst., 35 (2015), 2334–2352. doi: 10.1017/etds.2014.31. [16] J. Xu, K. Wang and M. Zhu, On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters, P. Am. Math. Soc., 144 (2016), 4793–4805. doi: 10.1090/proc/13088. [17] L. Zhang and J. Xu, Persistence of invariant tori in Hamiltonian systems with two-degree of freedom, J. Math. Anal. Appl., 338 (2008), 793-802. doi: 10.1016/j.jmaa.2007.05.052.

show all references

##### References:
 [1] N. N. Bogoljubov, J. A. Mitropoliski and A. M. Samoilenko, Methods of accelerated convergence in nonlinear mechanics, Springer-Verlag, New York, 1976. [2] L. H. Eliasson, Floquent solutions for the one-dimensional quasi-periodic Schrödinger equation, Commun. Math. Phys., 146 (1992), 447–482. doi: 10.1007/BF02097013. [3] H.-L. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems, J. Dyn. Differ. Equ., 20 (2008), 831–866. doi: 10.1007/s10884-008-9113-6. [4] R. A. Johnson and G. R. Sell, Smoothness of spectral subbundles and reducibility of quasiperodic linear differential systems, J. Dyn. Differ. Equ., 41 (1981), 262–288. doi: 10.1016/0022-0396(81)90062-0. [5] A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differ. Equations, 98 (1992), 111–124. doi: 10.1016/0022-0396(92)90107-X. [6] A. Jorba and C. Simó, On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704–1737. doi: 10.1137/S0036141094276913. [7] J. Li and C. Zhu, On the reducibility of a class of finitely differentiable quasi-periodic linear systems, J. Math. Anal. Appl., 413 (2014), 69–83. doi: 10.1016/j.jmaa.2013.10.077. [8] J. Li, C. Zhu and S. Chen, On the reducibility of a class of quasi-periodic Hamiltonian systems with small perturbation parameter near the equilibrium, Qual. Theory Dyn. Syst., 16 (2017), 127–147. doi: 10.1007/s12346-015-0164-x. [9] J. Pöschel, Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., 127 (1990), 351–393. doi: 10.1007/BF02096763. [10] H. Rüssmann, Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities, Ergodic Theor. Dyn. Syst., 24 (2004), 1787–1832. doi: 10.1017/S0143385703000774. [11] X. Wang and J. Xu, On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point, Nonlinear Anal., 69 (2008), 2318–2329. doi: 10.1016/j.na.2007.08.016. [12] H. Whitney, Analytical extensions of differentiable functions defined in closed sets, Trans. A. M. S., 36 (1934), 63–89. doi: 10.1090/S0002-9947-1934-1501735-3. [13] J. Xu and J. You, On reducibility of linear differential equations with almost-periodic coefficients, Chinese Ann. Math. A(in Chinese), 17 (1996), 607–616. [14] J. Xu, On the reducibility of a class of linear differential equations with quasiperiodic coefficients, Mathematika, 46 (1999), 443–451. doi: 10.1112/S0025579300007907. [15] J. Xu and X. Lu, On the reducibility of two-dimensional linear quasi-periodic systems with small parameter, Ergodic Theor. Dyn. Syst., 35 (2015), 2334–2352. doi: 10.1017/etds.2014.31. [16] J. Xu, K. Wang and M. Zhu, On the reducibility of 2-dimensional linear quasi-periodic systems with small parameters, P. Am. Math. Soc., 144 (2016), 4793–4805. doi: 10.1090/proc/13088. [17] L. Zhang and J. Xu, Persistence of invariant tori in Hamiltonian systems with two-degree of freedom, J. Math. Anal. Appl., 338 (2008), 793-802. doi: 10.1016/j.jmaa.2007.05.052.
 [1] Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevrey-smooth Hamiltonian systems under Rüssmann's non-degeneracy condition. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 635-655. doi: 10.3934/dcds.2006.16.635 [2] Xiaocai Wang, Junxiang Xu. Gevrey-smoothness of invariant tori for analytic reversible systems under Rüssmann's non-degeneracy condition. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 701-718. doi: 10.3934/dcds.2009.25.701 [3] Pedro J. Torres, Zhibo Cheng, Jingli Ren. Non-degeneracy and uniqueness of periodic solutions for $2n$-order differential equations. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 2155-2168. doi: 10.3934/dcds.2013.33.2155 [4] Lipeng Duan, Jun Yang. On the non-degeneracy of radial vortex solutions for a coupled Ginzburg-Landau system. Discrete and Continuous Dynamical Systems, 2021, 41 (10) : 4767-4790. doi: 10.3934/dcds.2021056 [5] Robert Magnus, Olivier Moschetta. The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy. Communications on Pure and Applied Analysis, 2012, 11 (2) : 587-626. doi: 10.3934/cpaa.2012.11.587 [6] Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure and Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027 [7] Morimichi Kawasaki, Ryuma Orita. Computation of annular capacity by Hamiltonian Floer theory of non-contractible periodic trajectories. Journal of Modern Dynamics, 2017, 11: 313-339. doi: 10.3934/jmd.2017013 [8] V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277 [9] Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024 [10] Paolo Perfetti. Hamiltonian equations on $\mathbb{T}^\infty$ and almost-periodic solutions. Conference Publications, 2001, 2001 (Special) : 303-309. doi: 10.3934/proc.2001.2001.303 [11] Sorin Micu, Ademir F. Pazoto. Almost periodic solutions for a weakly dissipated hybrid system. Mathematical Control and Related Fields, 2014, 4 (1) : 101-113. doi: 10.3934/mcrf.2014.4.101 [12] Zhichao Ma, Junxiang Xu. A KAM theorem for quasi-periodic non-twist mappings and its application. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022013 [13] Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9 [14] Fuzhong Cong, Yong Li. Invariant hyperbolic tori for Hamiltonian systems with degeneracy. Discrete and Continuous Dynamical Systems, 1997, 3 (3) : 371-382. doi: 10.3934/dcds.1997.3.371 [15] Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57 [16] Giuseppe Cordaro. Existence and location of periodic solutions to convex and non coercive Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 983-996. doi: 10.3934/dcds.2005.12.983 [17] César J. Niche. Non-contractible periodic orbits of Hamiltonian flows on twisted cotangent bundles. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 617-630. doi: 10.3934/dcds.2006.14.617 [18] Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053 [19] Shaoyun Shi, Wenlei Li. Non-integrability of generalized Yang-Mills Hamiltonian system. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1645-1655. doi: 10.3934/dcds.2013.33.1645 [20] Xiao Wang, Zhaohui Yang, Xiongwei Liu. Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6123-6138. doi: 10.3934/dcds.2017263

2020 Impact Factor: 1.327