On the reducibility of a class of almost periodic Hamiltonian systems

Jia Li; Junxiang Xu

doi:10.3934/dcdsb.2020268

Article Contents

2021, Volume 26, Issue 7: 3905-3919. Doi: 10.3934/dcdsb.2020268

This issue Previous Article A class of stochastic Fredholm-algebraic equations and applications in finance Next Article Numerical analysis of two new finite difference methods for time-fractional telegraph equation

On the reducibility of a class of almost periodic Hamiltonian systems

Jia Li^a, and
Junxiang Xu^b,

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

Early access: September 2020

Published: July 2021

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: 37J40.

Citation:

Full Text(HTML)

Related Papers

Cited by

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.