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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 and Applied Analysis, 2015, 14 (2) : 421-437. doi: 10.3934/cpaa.2015.14.421
##### References:
 [1] N. N. Bogoljubov, Y. A. Mitropolskii and A. M. Samoilenko, Methods Of Accelerated Convergence In Nonlinear Mechanics, Springer, New York, 1976 (Russian original: Naukova Dumka, Kiev 1969). [2] H. Her and J. You, Full measure reducibility for generic one-parameter family of quasiperiodic linear systems, J. Dynam. Differential Equations, 20 (2008), 831-866. doi: 10.1007/s10884-008-9113-6. [3] 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. [4] A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X. [5] J. Moser, Convergent series expansion for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. [6] J. Xu, On Quasi-periodic perturbations of hyperbalic-type degeneate equilibrium point of a class of planar system, Discrete and Continous Dynamical Systems, 33 (2013), 2593-2619. doi: doi:10.3934/dcds.2013.33.2593. [7] J. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations, 250 (2011), 551-571. doi: 10.1016/j.jde.2010.09.030. [8] J. Xu, Persistence of Floquet invariant tori for a class of non-conservative dynamical systems, Proc. Amer. Math. Soc., 135 (2007), 805-814. doi: 10.1090/S0002-9939-06-08529-7. [9] J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate, Proc. Amer. Math. Soc., 126 (1998), 1445-1451. doi: 10.1090/S0002-9939-98-04523-7. [10] J. Xu and S. Jiang, Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation, Ergodic Theory and Dynamical Systems, 31 (2011), 599-611. doi: 10.1017/S0143385709001114.

show all references

##### References:
 [1] N. N. Bogoljubov, Y. A. Mitropolskii and A. M. Samoilenko, Methods Of Accelerated Convergence In Nonlinear Mechanics, Springer, New York, 1976 (Russian original: Naukova Dumka, Kiev 1969). [2] H. Her and J. You, Full measure reducibility for generic one-parameter family of quasiperiodic linear systems, J. Dynam. Differential Equations, 20 (2008), 831-866. doi: 10.1007/s10884-008-9113-6. [3] 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. [4] A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients, J. Differential Equations, 98 (1992), 111-124. doi: 10.1016/0022-0396(92)90107-X. [5] J. Moser, Convergent series expansion for quasi-periodic motions, Math. Ann., 169 (1967), 136-176. [6] J. Xu, On Quasi-periodic perturbations of hyperbalic-type degeneate equilibrium point of a class of planar system, Discrete and Continous Dynamical Systems, 33 (2013), 2593-2619. doi: doi:10.3934/dcds.2013.33.2593. [7] J. Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differential Equations, 250 (2011), 551-571. doi: 10.1016/j.jde.2010.09.030. [8] J. Xu, Persistence of Floquet invariant tori for a class of non-conservative dynamical systems, Proc. Amer. Math. Soc., 135 (2007), 805-814. doi: 10.1090/S0002-9939-06-08529-7. [9] J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate, Proc. Amer. Math. Soc., 126 (1998), 1445-1451. doi: 10.1090/S0002-9939-98-04523-7. [10] J. Xu and S. Jiang, Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation, Ergodic Theory and Dynamical Systems, 31 (2011), 599-611. doi: 10.1017/S0143385709001114.
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