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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

Junxiang Xu 1,

1. 

Department of Mathematics, Southeast University, Nanjing 210096

Received  November 2011 Revised  October 2012 Published  December 2012

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, 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), viii+175.  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), 385-419. doi: 10.1007/s002200050618.  Google Scholar

[4]

L. H. Eliasson, Almost reducibility of linear quasi-periodic systems, Smooth ergodic theory and its applications (Seattle, WA, 1999), 679-705, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, (2001).  Google Scholar

[5]

S. M. Graff, On the conservation of hyperbolic invariant tori for hamiltonian systems, J. Differential Equations, 15 (1974), 1-69.  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), 831-866. 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), 111-124. 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), 1704-1737. doi: 10.1137/S0036141094276913.  Google Scholar

[9]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1976), 136-176.  Google Scholar

[10]

J. Pöschel, On elliptic lower dimensional tori in hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.  Google Scholar

[11]

W. Rudin, "Real and Complex Analysis," Third Edition, McGraw-Hill Compnies, Inc., 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), 1445-1451. 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), 805-814. 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), 599-611. 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), 551-571. 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), 145-168. doi: 10.1007/s002200050294.  Google Scholar

show all references

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), viii+175.  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), 385-419. doi: 10.1007/s002200050618.  Google Scholar

[4]

L. H. Eliasson, Almost reducibility of linear quasi-periodic systems, Smooth ergodic theory and its applications (Seattle, WA, 1999), 679-705, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, (2001).  Google Scholar

[5]

S. M. Graff, On the conservation of hyperbolic invariant tori for hamiltonian systems, J. Differential Equations, 15 (1974), 1-69.  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), 831-866. 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), 111-124. 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), 1704-1737. doi: 10.1137/S0036141094276913.  Google Scholar

[9]

J. Moser, Convergent series expansions for quasi-periodic motions, Math. Ann., 169 (1976), 136-176.  Google Scholar

[10]

J. Pöschel, On elliptic lower dimensional tori in hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.  Google Scholar

[11]

W. Rudin, "Real and Complex Analysis," Third Edition, McGraw-Hill Compnies, Inc., 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), 1445-1451. 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), 805-814. 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), 599-611. 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), 551-571. 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), 145-168. doi: 10.1007/s002200050294.  Google Scholar

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