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On the reducibility of two-dimensional linear quasi-periodic systems with small parameter

Published online by Cambridge University Press:  04 August 2014

JUNXIANG XU  and
XUEZHU LU
JUNXIANG XU
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, PR China email xujun@seu.edu.cn, jennysundate@gmail.com
XUEZHU LU
Affiliation:
Department of Mathematics, Southeast University, Nanjing 210096, PR China email xujun@seu.edu.cn, jennysundate@gmail.com
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Abstract

In this paper we consider a linear real analytic quasi-periodic system of two differential equations, whose coefficient matrix analytically depends on a small parameter and closes to constant. Under some non-resonance conditions about the basic frequencies and the eigenvalues of the constant matrix and without any non-degeneracy assumption of the small parameter, we prove that the system is reducible for most of the sufficiently small parameters in the sense of the Lebesgue measure.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 35 , Issue 7 , October 2015 , pp. 2334 - 2352
Copyright
© Cambridge University Press, 2014 

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References

Bogoljubov, N. N., Mitropol’ski, J. A. and Samoilenko, A. M.. Methods of Accelerated Convergence in Nonlinear Mechanics. Springer, New York, 1976.CrossRefGoogle Scholar
Broer, H. W., Huitema, G. B. and Sevryuk, M. B.. Quasi-periodic Motions in Famillies of Dynamical Systems (Lecture Notes in Mathematics, 1645). Springer, 1996.Google Scholar
Dinaburg, E. I. and Sinai, Ya. G.. The one-dimensional Schrödinger equation with a quasi-periodic potential. Funct. Anal. Appl. 9 (1975), 279–289.CrossRefGoogle Scholar
Eliasson, L. H.. Floquet solutions for the one-dimensional Schrödinger quasiperiodic equations. Comm. Math. Phys. 146 (1992), 447–482.CrossRefGoogle Scholar
Eliasson, L. H.. Almost reducibility of linear quasi-periodic systems. Proc. Sympos. Pure Math. 69 (2001), 679–705.CrossRefGoogle Scholar
Eliasson, L. H. and Kuksin, S. B.. On reducibility of Schrödinger equations with quasiperiodic in time potential. Comm. Math. Phys. 286 (2009), 125–135.CrossRefGoogle Scholar
Fayad, B. and Krikorian, R.. Herman’s last geometric theorem. Ann. Sci. École Norm. Supér. 42(2) (2009), 193–219.CrossRefGoogle Scholar
Her, H. and You, J.. Full measure reducibility for generic one-parameter family of quasiperiodic linear systems. J. Dynam. Differential Equations 20 (2008), 831–866.CrossRefGoogle Scholar
Hou, X. and You, J.. Almost reducibility and non-perturbative reducibility of quasi-periodic linear systems. Invent. Math. 190 (2012), 209–260.CrossRefGoogle Scholar
Johnson, R. A. and Sell, G. R.. Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems. J. Differential Equations 41 (1981), 262–288.CrossRefGoogle Scholar
Jorba, A. and Simó, C.. On the reducibility of linear differential equations with quasi-periodic coefficients. J. Differential Equations 98 (1992), 111–124.CrossRefGoogle Scholar
Jorba, A. and Simó, C.. On quasi-periodic perturbations of elliptic equilibrium points. SIAM J. Math. Anal. 27(6) (1996), 1704–1737.CrossRefGoogle Scholar
Johnson, R. and Moser, J. . The rotation number for almost periodic potentials. Comm. Math. Phys. 84 ( 1982), 403–438.CrossRefGoogle Scholar
Rüssmann, H.. On the one-dimensional Schrödinger equation with a quasi-periodic potential. Ann. New York Acad. Sci. 357 (1980), 90–107.CrossRefGoogle Scholar
Rüssmann, H.. Convergent transformations into a normal form in analytic Hamiltonian systems with two degrees of freedom on the zero energy surface near degenerate elliptic singularities. Ergod. Th. & Dynam. Sys. 24 (2004), 1787–1832.CrossRefGoogle Scholar
Whitney, H.. Analytic extensions of differentiable functions defined in closed sets. Trans. Amer. Math. Soc. 36 (1934), 63–89.CrossRefGoogle Scholar
Xu, J. and Zheng, Q. . On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate. Proc. Amer. Math. Soc. 126(5) (1998), 1445–1451.Google Scholar