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November  2020, 19(11): 5157-5180. doi: 10.3934/cpaa.2020231

Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations

Jinjing Jiao and  Guanghua Shi ,

College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410006, China

* Corresponding author

Received  November 2019 Revised  May 2020 Published  November 2020 Early access  September 2020

Fund Project: The second author is supported by the Education Department Project of Hunan Province (No.18C0026) and the NNSF of China (No.11971163)

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.

Keywords: KAM theory, quasi-periodic solution, elliptic degenerate equilibrium point.
Mathematics Subject Classification: Primary: 34J40, 34C27; Secondary: 34E20.
Citation: Jinjing Jiao, Guanghua Shi. Quasi-periodic solutions for the two-dimensional systems with an elliptic-type degenerate equilibrium point under small perturbations. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5157-5180. doi: 10.3934/cpaa.2020231
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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