On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point

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March 2015, 14(2): 421-437. doi: 10.3934/cpaa.2015.14.421

On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point

1.	School of Mathematics, Shandong University, Jinan, Shandong 250100, China, China

Received December 2013 Revised July 2014 Published December 2014

Abstract
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This paper focuses on quasi-periodic perturbation of four dimensional nonlinear quasi-periodic system. Using the KAM method, the perturbed system can be reduced to a suitable normal form with zero as equilibrium point by a quasi-periodic transformation. Hence, the perturbed system has a quasi-periodic solution near the equilibrium point.

Keywords: normal form, small perturbation, degenerate equilibrium point., KAM method, Quasi-periodic solution.

Mathematics Subject Classification: Primary: 34C20; Secondary: 34D1.

Citation: Wenhua Qiu, Jianguo Si. On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point. Communications on Pure & Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421

References:

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References:

[1]	N. N. Bogoljubov, Y. A. Mitropolskii and A. M. Samoilenko, Methods Of Accelerated Convergence In Nonlinear Mechanics,, Springer, (1976). Google Scholar
[2]	H. Her and J. You, Full measure reducibility for generic one-parameter family of quasiperiodic linear systems,, \emph{J. Dynam. Differential Equations}, 20 (2008), 831. doi: 10.1007/s10884-008-9113-6. Google Scholar
[3]	A. Jorba and C. Simó, On quasi-periodic perturbations of elliptic equilibrium points,, \emph{SIAM J. Math. Anal.}, 27 (1996), 1704. doi: 10.1137/S0036141094276913. Google Scholar
[4]	A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients,, \emph{J. Differential Equations}, 98 (1992), 111. doi: 10.1016/0022-0396(92)90107-X. Google Scholar
[5]	J. Moser, Convergent series expansion for quasi-periodic motions,, \emph{Math. Ann.}, 169 (1967), 136. Google Scholar
[6]	J. Xu, On Quasi-periodic perturbations of hyperbalic-type degeneate equilibrium point of a class of planar system,, \emph{Discrete and Continous Dynamical Systems}, 33 (2013), 2593. doi: doi:10.3934/dcds.2013.33.2593. Google Scholar
[7]	J. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point,, \emph{J. Differential Equations}, 250 (2011), 551. doi: 10.1016/j.jde.2010.09.030. Google Scholar
[8]	J. Xu, Persistence of Floquet invariant tori for a class of non-conservative dynamical systems,, \emph{Proc. Amer. Math. Soc.}, 135 (2007), 805. doi: 10.1090/S0002-9939-06-08529-7. Google Scholar
[9]	J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate,, \emph{Proc. Amer. Math. Soc.}, 126 (1998), 1445. doi: 10.1090/S0002-9939-98-04523-7. Google Scholar
[10]	J. Xu and S. Jiang, Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation,, \emph{Ergodic Theory and Dynamical Systems}, 31 (2011), 599. doi: 10.1017/S0143385709001114. Google Scholar

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