On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems

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June 2013, 33(6): 2593-2619. doi: 10.3934/dcds.2013.33.2593

On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems

1.	Department of Mathematics, Southeast University, Nanjing 210096

Received November 2011 Revised October 2012 Published December 2012

Abstract
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In this paper we consider two-dimensional nonlinear quasi-periodic system with small perturbations. Assume that the unperturbed system has a hyperbolic-type degenerate equilibrium point and the frequency satisfies the Diophantine conditions. Using the KAM iteration we prove that for sufficiently small perturbations, the system can be reduced by a nonlinear quasi-periodic transformation to a suitable normal form with an equilibrium point at the origin. Hence, for the system we can obtain a small quasi-periodic solution.

Keywords: Diophantine conditions, KAM-iteration, Quasi-periodic systems, hyperbolic equilibrium point..

Mathematics Subject Classification: Primary: 34J40, 34C27; Secondary: 34E2.

Citation: Junxiang Xu. On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2593-2619. doi: 10.3934/dcds.2013.33.2593

References:

[1]	H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori,, Mem. Amer. Math. Soc., 83 (1990). Google Scholar
[2]	H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-periodic Motions in Families of Dynamical Systems,", Lecture Notes in Mathematics, 1645 (1996). Google Scholar
[3]	C. Q. Cheng, Lower diemsional invariant tori in the regions of instability of nearly integrable hamiltonian systems,, Commun. Math. Phys., 203 (1999), 385. doi: 10.1007/s002200050618. Google Scholar
[4]	L. H. Eliasson, Almost reducibility of linear quasi-periodic systems,, Smooth ergodic theory and its applications (Seattle, 69 (2001), 679. Google Scholar
[5]	S. M. Graff, On the conservation of hyperbolic invariant tori for hamiltonian systems,, J. Differential Equations, 15 (1974), 1. Google Scholar
[6]	H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems,, J. Dynam. Differential Equations, 20 (2008), 831. doi: 10.1007/s10884-008-9113-6. Google Scholar
[7]	A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients,, J. Differential Equations, 98 (1992), 111. doi: 10.1016/0022-0396(92)90107-X. Google Scholar
[8]	A. Jorba and C. Simó, On quasi-periodic perturbations of elliptic equilibrium points,, SIAM J. Math. Anal., 27 (1996), 1704. doi: 10.1137/S0036141094276913. Google Scholar
[9]	J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1976), 136. Google Scholar
[10]	J. Pöschel, On elliptic lower dimensional tori in hamiltonian systems,, Math. Z., 202 (1989), 559. doi: 10.1007/BF01221590. Google Scholar
[11]	W. Rudin, "Real and Complex Analysis,", Third Edition, (2003). Google Scholar
[12]	Junxiang Xu and Qin Zheng, On the reducibility of linear differential equations with quasi-periodic coefficients which are degenerate,, Proc. Amer. Math. Soc., 126 (1998), 1445. doi: 10.1090/S0002-9939-98-04523-7. Google Scholar
[13]	Junxiang Xu, Persistence of Floquet invariant tori for a class of non-conservative dynamical systems,, Proc. Amer. Math. Soc., 135 (2007), 805. doi: 10.1090/S0002-9939-06-08529-7. Google Scholar
[14]	Junxiang Xu and Shunjun 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. doi: 10.1017/S0143385709001114. Google Scholar
[15]	Junxiang Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point,, J. Differential Equations, 250 (2011), 551. doi: 10.1016/j.jde.2010.09.030. Google Scholar
[16]	J. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems,, Commun. Math. Phys., 192 (1998), 145. doi: 10.1007/s002200050294. Google Scholar

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References:

[1]	H. W. Broer, G. B. Huitema, F. Takens and B. L. J. Braaksma, Unfoldings and bifurcations of quasi-periodic tori,, Mem. Amer. Math. Soc., 83 (1990). Google Scholar
[2]	H. W. Broer, G. B. Huitema and M. B. Sevryuk, "Quasi-periodic Motions in Families of Dynamical Systems,", Lecture Notes in Mathematics, 1645 (1996). Google Scholar
[3]	C. Q. Cheng, Lower diemsional invariant tori in the regions of instability of nearly integrable hamiltonian systems,, Commun. Math. Phys., 203 (1999), 385. doi: 10.1007/s002200050618. Google Scholar
[4]	L. H. Eliasson, Almost reducibility of linear quasi-periodic systems,, Smooth ergodic theory and its applications (Seattle, 69 (2001), 679. Google Scholar
[5]	S. M. Graff, On the conservation of hyperbolic invariant tori for hamiltonian systems,, J. Differential Equations, 15 (1974), 1. Google Scholar
[6]	H. Her and J. You, Full measure reducibility for generic one-parameter family of quasi-periodic linear systems,, J. Dynam. Differential Equations, 20 (2008), 831. doi: 10.1007/s10884-008-9113-6. Google Scholar
[7]	A. Jorba and C. Simó, On the reducibility of linear differential equations with quasiperiodic coefficients,, J. Differential Equations, 98 (1992), 111. doi: 10.1016/0022-0396(92)90107-X. Google Scholar
[8]	A. Jorba and C. Simó, On quasi-periodic perturbations of elliptic equilibrium points,, SIAM J. Math. Anal., 27 (1996), 1704. doi: 10.1137/S0036141094276913. Google Scholar
[9]	J. Moser, Convergent series expansions for quasi-periodic motions,, Math. Ann., 169 (1976), 136. Google Scholar
[10]	J. Pöschel, On elliptic lower dimensional tori in hamiltonian systems,, Math. Z., 202 (1989), 559. doi: 10.1007/BF01221590. Google Scholar
[11]	W. Rudin, "Real and Complex Analysis,", Third Edition, (2003). Google Scholar
[12]	Junxiang Xu and Qin Zheng, On the reducibility of linear differential equations with quasi-periodic coefficients which are degenerate,, Proc. Amer. Math. Soc., 126 (1998), 1445. doi: 10.1090/S0002-9939-98-04523-7. Google Scholar
[13]	Junxiang Xu, Persistence of Floquet invariant tori for a class of non-conservative dynamical systems,, Proc. Amer. Math. Soc., 135 (2007), 805. doi: 10.1090/S0002-9939-06-08529-7. Google Scholar
[14]	Junxiang Xu and Shunjun 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. doi: 10.1017/S0143385709001114. Google Scholar
[15]	Junxiang Xu, On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point,, J. Differential Equations, 250 (2011), 551. doi: 10.1016/j.jde.2010.09.030. Google Scholar
[16]	J. You, A KAM theorem for hyperbolic-type degenerate lower dimensional tori in Hamiltonian systems,, Commun. Math. Phys., 192 (1998), 145. doi: 10.1007/s002200050294. Google Scholar

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