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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

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.
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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show all references

##### 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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