American Institute of Mathematical Sciences

Advanced
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 - 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

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).   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

[1]

Koichiro Naito. Recurrent dimensions of quasi-periodic solutions for nonlinear evolution equations II: Gaps of dimensions and Diophantine conditions. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 449-488. doi: 10.3934/dcds.2004.11.449
[2]

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
[3]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[4]

Qihuai Liu, Dingbian Qian, Zhiguo Wang. Quasi-periodic solutions of the Lotka-Volterra competition systems with quasi-periodic perturbations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1537-1550. doi: 10.3934/dcdsb.2012.17.1537
[5]

Helmut Rüssmann. KAM iteration with nearly infinitely small steps in dynamical systems of polynomial character. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 683-718. doi: 10.3934/dcdss.2010.3.683
[6]

Wen Si, Fenfen Wang, Jianguo Si. Almost-periodic perturbations of non-hyperbolic equilibrium points via Pöschel-Rüssmann KAM method. Communications on Pure & Applied Analysis, 2020, 19 (1) : 541-585. doi: 10.3934/cpaa.2020027
[7]

Ernest Fontich, Rafael de la Llave, Yannick Sire. A method for the study of whiskered quasi-periodic and almost-periodic solutions in finite and infinite dimensional Hamiltonian systems. Electronic Research Announcements, 2009, 16: 9-22. doi: 10.3934/era.2009.16.9
[8]

Jean Bourgain. On quasi-periodic lattice Schrödinger operators. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 75-88. doi: 10.3934/dcds.2004.10.75
[9]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
[10]

Mikhail B. Sevryuk. Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 569-595. doi: 10.3934/dcds.2007.18.569
[11]

Yanling Shi, Junxiang Xu, Xindong Xu. Quasi-periodic solutions of generalized Boussinesq equation with quasi-periodic forcing. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2501-2519. doi: 10.3934/dcdsb.2017104
[12]

Lei Jiao, Yiqian Wang. The construction of quasi-periodic solutions of quasi-periodic forced Schrödinger equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1585-1606. doi: 10.3934/cpaa.2009.8.1585
[13]

Alessandro Fonda, Antonio J. Ureña. Periodic, subharmonic, and quasi-periodic oscillations under the action of a central force. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 169-192. doi: 10.3934/dcds.2011.29.169
[14]

Xavier Blanc, Claude Le Bris. Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Networks & Heterogeneous Media, 2010, 5 (1) : 1-29. doi: 10.3934/nhm.2010.5.1
[15]

Russell Johnson, Francesca Mantellini. A nonautonomous transcritical bifurcation problem with an application to quasi-periodic bubbles. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 209-224. doi: 10.3934/dcds.2003.9.209
[16]

Siqi Xu, Dongfeng Yan. Smooth quasi-periodic solutions for the perturbed mKdV equation. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1857-1869. doi: 10.3934/cpaa.2016019
[17]

Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[18]

Meina Gao, Jianjun Liu. Quasi-periodic solutions for derivative nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[19]

Xuanji Hou, Lei Jiao. On local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3125-3152. doi: 10.3934/dcds.2016.36.3125
[20]

Zhenguo Liang, Jiansheng Geng. Quasi-periodic solutions for 1D resonant beam equation. Communications on Pure & Applied Analysis, 2006, 5 (4) : 839-853. doi: 10.3934/cpaa.2006.5.839

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (15)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences