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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

Wenhua Qiu 1, and  Jianguo Si 1,

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.
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 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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