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Uniform stabilization of the Klein-Gordon system
Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations
College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410006, China |
In this paper, we consider the existence of quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. We prove that under appropriate hypotheses there exist quasi-periodic solutions for perturbed ODEs near the equilibrium point for most parameter values.
References:
[1] |
V. I. Arnold,
Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192.
|
[2] |
J. L. Braaksma and H. W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. Hemri Poincaré Anal. Nonlinear, 4 (1987), 115–168. |
[3] |
J. L. Braaksma, H. W. Broer and G. B. Huitema,
Toward a quasi-periodic bifurcation theory, Mem. Am. Math. Soc., 83 (1990), 83-167.
|
[4] |
H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems: Order Amidst Chaos, Lecture Notes in Math, Springer, Berlin, 1996. |
[5] |
M. Friedman,
Quasi-periodic solutions of nonlinear ordinary differential equations with small damping, Bull. Am. Math. Soc., 73 (1967), 460-464.
doi: 10.1090/S0002-9904-1967-11783-X. |
[6] |
G. Gentile,
Quasi-periodic motions in strongly dissipative forced systems, Ergod. Theor. Dyn. Syst., 30 (2010), 1457-1469.
doi: 10.1017/S0143385709000583. |
[7] |
A. Jorba and C. Simo,
On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737.
doi: 10.1137/S0036141094276913. |
[8] |
X. Li and R. de la Llave,
Construction of quasi-periodic solutions of delay differential equations via KAM technique, J. Differ. Equ., 247 (2009), 822-865.
doi: 10.1016/j.jde.2009.03.009. |
[9] |
X. Li and X. Yuan,
Quasi-periodic solution for perturbed autonomous delay differential equations, J. Differ. Equ., 252 (2012), 3752-3796.
doi: 10.1016/j.jde.2011.11.014. |
[10] |
X. Li and Z. Shang,
Quasi-Periodic Solutions for Differential Equations with an Elliptic-Type Degenerate Equilibrium Point Under Small Perturbations, J. Dyn. Differ. Equ., 31 (2019), 653-681.
doi: 10.1007/s10884-018-9642-6. |
[11] |
J. Moser,
Combiantion tones for Duffing's equation, Commun. Pure Appl. Math., 18 (1965), 167-181.
doi: 10.1002/cpa.3160180116. |
[12] |
J. Moser,
Convergent series expensions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.
doi: 10.1007/BF01399536. |
[13] |
J. Pöschel,
A lecture on the classical KAM theorem, Proc. Symp. Pure Math., 69 (2001), 707-732.
doi: 10.1090/pspum/069/1858551. |
[14] |
W. Qiu and J. Si,
On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point, Comm. Pure Appl. Math., 14 (2015), 421-437.
doi: 10.3934/cpaa.2015.14.421. |
[15] |
M. B. Sevryuk,
Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method, Discrete Contin. Dyn. Syst., 18 (2007), 569-595.
doi: 10.3934/dcds.2007.18.569. |
[16] |
W. Si and J. Si,
Elliptic-type degenerate invariant tori for quasi-periodically forced four-dimensional non-conservative systems, J. Math. Anal. Appl., 460 (2018), 164-202.
doi: 10.1016/j.jmaa.2017.11.047. |
[17] |
W. Si and J. Si,
Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrum point under small perturbations, J. Differ. Equ., 262 (2017), 4771-4822.
doi: 10.1016/j.jde.2016.12.019. |
[18] |
W. Si and J. Si,
Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems, Nonlinearity, 31 (2018), 2361-2418.
doi: 10.1088/1361-6544/aaa7b9. |
[19] |
W. Si, F. Wang and J. Si,
Almost-periodic perturbations of nonhyperbolic equilibrium points via P$\ddot{o}$schel-R$\ddot{u}$ssmann KAM method, Commun. Pure Appl. Anal., 19 (2020), 541-585.
doi: 10.3934/cpaa.2020027. |
[20] |
J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Wiley, 1992, Interscience, New York, 1950. |
[21] |
J. Xu,
On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differ. Equ., 250 (2011), 551-571.
doi: 10.1016/j.jde.2010.09.030. |
[22] |
J. Xu,
On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems, Discrete Contin. Dyn. Syst., 33 (2013), 2593-2619.
doi: 10.3934/dcds.2013.33.2593. |
[23] |
J. Xu and S. Jiang,
Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation, Ergod. Theor. Dyn. Syst., 31 (2011), 599-611.
doi: 10.1017/S0143385709001114. |
[24] |
J. Xu and J. You,
Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition, J. Math. Pures Appl., 9 (2001), 1045-1067.
doi: 10.1016/S0021-7824(01)01221-1. |
[25] |
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. |
[26] |
X. Yuan,
Construction of quasi-periodic breathers via KAM technique, Commun. Math. Phys., 226 (2002), 61-100.
doi: 10.1007/s002200100593. |
show all references
References:
[1] |
V. I. Arnold,
Small denominators and problems of stability of motion in classical and celestial mechanics, Uspehi Mat. Nauk, 18 (1963), 91-192.
|
[2] |
J. L. Braaksma and H. W. Broer, On a quasi-periodic Hopf bifurcation, Ann. Inst. Hemri Poincaré Anal. Nonlinear, 4 (1987), 115–168. |
[3] |
J. L. Braaksma, H. W. Broer and G. B. Huitema,
Toward a quasi-periodic bifurcation theory, Mem. Am. Math. Soc., 83 (1990), 83-167.
|
[4] |
H. W. Broer, G. B. Huitema and M. B. Sevryuk, Quasi-Periodic Motions in Families of Dynamical Systems: Order Amidst Chaos, Lecture Notes in Math, Springer, Berlin, 1996. |
[5] |
M. Friedman,
Quasi-periodic solutions of nonlinear ordinary differential equations with small damping, Bull. Am. Math. Soc., 73 (1967), 460-464.
doi: 10.1090/S0002-9904-1967-11783-X. |
[6] |
G. Gentile,
Quasi-periodic motions in strongly dissipative forced systems, Ergod. Theor. Dyn. Syst., 30 (2010), 1457-1469.
doi: 10.1017/S0143385709000583. |
[7] |
A. Jorba and C. Simo,
On quasi-periodic perturbations of elliptic equilibrium points, SIAM J. Math. Anal., 27 (1996), 1704-1737.
doi: 10.1137/S0036141094276913. |
[8] |
X. Li and R. de la Llave,
Construction of quasi-periodic solutions of delay differential equations via KAM technique, J. Differ. Equ., 247 (2009), 822-865.
doi: 10.1016/j.jde.2009.03.009. |
[9] |
X. Li and X. Yuan,
Quasi-periodic solution for perturbed autonomous delay differential equations, J. Differ. Equ., 252 (2012), 3752-3796.
doi: 10.1016/j.jde.2011.11.014. |
[10] |
X. Li and Z. Shang,
Quasi-Periodic Solutions for Differential Equations with an Elliptic-Type Degenerate Equilibrium Point Under Small Perturbations, J. Dyn. Differ. Equ., 31 (2019), 653-681.
doi: 10.1007/s10884-018-9642-6. |
[11] |
J. Moser,
Combiantion tones for Duffing's equation, Commun. Pure Appl. Math., 18 (1965), 167-181.
doi: 10.1002/cpa.3160180116. |
[12] |
J. Moser,
Convergent series expensions for quasi-periodic motions, Math. Ann., 169 (1967), 136-176.
doi: 10.1007/BF01399536. |
[13] |
J. Pöschel,
A lecture on the classical KAM theorem, Proc. Symp. Pure Math., 69 (2001), 707-732.
doi: 10.1090/pspum/069/1858551. |
[14] |
W. Qiu and J. Si,
On small perturbation of four-dimensional quasi-periodic system with degenerate equilibrium point, Comm. Pure Appl. Math., 14 (2015), 421-437.
doi: 10.3934/cpaa.2015.14.421. |
[15] |
M. B. Sevryuk,
Invariant tori in quasi-periodic non-autonomous dynamical systems via Herman's method, Discrete Contin. Dyn. Syst., 18 (2007), 569-595.
doi: 10.3934/dcds.2007.18.569. |
[16] |
W. Si and J. Si,
Elliptic-type degenerate invariant tori for quasi-periodically forced four-dimensional non-conservative systems, J. Math. Anal. Appl., 460 (2018), 164-202.
doi: 10.1016/j.jmaa.2017.11.047. |
[17] |
W. Si and J. Si,
Construction of response solutions for two classes of quasi-periodically forced four-dimensional nonlinear systems with degenerate equilibrum point under small perturbations, J. Differ. Equ., 262 (2017), 4771-4822.
doi: 10.1016/j.jde.2016.12.019. |
[18] |
W. Si and J. Si,
Response solutions and quasi-periodic degenerate bifurcations for quasi-periodically forced systems, Nonlinearity, 31 (2018), 2361-2418.
doi: 10.1088/1361-6544/aaa7b9. |
[19] |
W. Si, F. Wang and J. Si,
Almost-periodic perturbations of nonhyperbolic equilibrium points via P$\ddot{o}$schel-R$\ddot{u}$ssmann KAM method, Commun. Pure Appl. Anal., 19 (2020), 541-585.
doi: 10.3934/cpaa.2020027. |
[20] |
J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, Wiley, 1992, Interscience, New York, 1950. |
[21] |
J. Xu,
On small perturbation of two-dimensional quasi-periodic systems with hyperbolic-type degenerate equilibrium point, J. Differ. Equ., 250 (2011), 551-571.
doi: 10.1016/j.jde.2010.09.030. |
[22] |
J. Xu,
On quasi-periodic perturbations of hyperbolic-type degenerate equilibrium point of a class of planar systems, Discrete Contin. Dyn. Syst., 33 (2013), 2593-2619.
doi: 10.3934/dcds.2013.33.2593. |
[23] |
J. Xu and S. Jiang,
Reducibility for a class of nonlinear quasi-periodic differential equations with degenerate equilibrium point under small perturbation, Ergod. Theor. Dyn. Syst., 31 (2011), 599-611.
doi: 10.1017/S0143385709001114. |
[24] |
J. Xu and J. You,
Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition, J. Math. Pures Appl., 9 (2001), 1045-1067.
doi: 10.1016/S0021-7824(01)01221-1. |
[25] |
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. |
[26] |
X. Yuan,
Construction of quasi-periodic breathers via KAM technique, Commun. Math. Phys., 226 (2002), 61-100.
doi: 10.1007/s002200100593. |
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