American Institute of Mathematical Sciences

Advanced
June  2017, 10(3): 543-556. doi: 10.3934/dcdss.2017027

Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term

Xiaoming Wang 1,2,,

1. 

School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China
2. 

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China

* Corresponding author

Received  December 2015 Revised  October 2016 Published  February 2017

Fund Project: The work is supported by NSF of China under 11461056.

In this paper we consider the existence of Aubry-Mather sets and quasi-periodic solutions for a class of second order differential equation with a nonlinear damping term
$$x''+α x^+-β x^-+q(x)f(x')+g(t,x)=p(t), $$
where
$q, f∈ C^1(\mathbb{R}),$
$g(t,x)∈ C^{0,1}(\mathbf{S}^1× \mathbb{R})$
and
$p(t)∈ C^0(\mathbf{S}^1)$
,
$\mathbf{S}^1= \mathbb{R}/2π\mathbb{Z}$
,
$α$
and
$β $
are two positive constants satisfying
$$\frac{1}{\sqrt{α}}+\frac{1}{\sqrt{β}}=\frac{2}{ω}$$
with
$ω∈ \mathbb{R}^+ $
. Under some assumptions on the parities of
$f,$
$g$
and
$p$
, we obtain the existence of infinitely many generalized quasi-periodic solutions via a result of Chow and Pei from the Aubry-Mather theory of reversible mapping. In particular, an advantage of our approach is that it does not require any high smoothness assumptions on the functions
$q, f, g$
and
$p$
.
Keywords: Reversible systems, asymmetric oscillator, quasi-periodic solutions, Aubry-Mather sets.
Mathematics Subject Classification: Primary: 34C12; Secondary: 37C55.
Citation: Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
References:
[1]

A. Capietto and B. Liu, Quasi-periodic solutions of a forced asymmetric oscillator at resonance, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 105-117.  doi: 10.1016/j.na.2003.09.001.  Google Scholar

[2]

A. Capietto, W. Dambrosio and B. Liu, On the boundedness of solutions to a nonlinear singular oscillator, Z. Angew. Math. Phys., 60 (2009), 1007-1034.  doi: 10.1007/s00033-008-8094-y.  Google Scholar

[3]

A. Capietto, W. Dambrosio and X. Wang, Quasi-periodic solutions of a damped reversible oscillator at resonance, Differential and Integral Equations, 22 (2009), 1033-1046.   Google Scholar

[4]

S. Chow and M. Pei, Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems, Nonlinear Analysis: Theory, Methods} & \emph{Applications, 25 (1995), 905-931.  doi: 10.1016/0362-546X(95)00087-C.  Google Scholar

[5]

B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.  doi: 10.1006/jmaa.1998.6219.  Google Scholar

[6]

B. Liu and J. Song, Invariant curved of reversible mappings with small twist, Acta Math. Sinica (English Series), 20 (2004), 15-24.  doi: 10.1007/s10114-004-0316-4.  Google Scholar

[7]

X. Li, Invariant tori for semilinear reversible systems, Nonlinear Analysis, 56 (2004), 133-146.  doi: 10.1016/j.na.2003.09.004.  Google Scholar

[8]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.  Google Scholar

[9]

M. Pei, Mather sets for superlinear Duffing equations, Science in China, Ser. A, 36 (1993), 524-537.   Google Scholar

[10]

M. Pei, Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations, 113 (1994), 106-127.  doi: 10.1006/jdeq.1994.1116.  Google Scholar

[11]

D. Qian, Mather sets for sublinear Duffing Equations, Chin. Ann. of Math., Ser. B, 15 (1994), 421-434.   Google Scholar

[12]

J. Si, Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term, Nonlinear Anal., 64 (2006), 1475-1495.  doi: 10.1016/j.na.2005.06.046.  Google Scholar

[13]

X. Wang, Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(English Series), 19 (2003), 765-782.  doi: 10.1007/s10114-003-0249-3.  Google Scholar

[14]

X. Wang, Aubry-Mather sets for semilinear Duffing equations, Acta Mathematica Sinica(Chinese Series), 52 (2009), 605-610.   Google Scholar

[15]

X. Wang, Aubry-Mather sets for sublinear asymmetric Duffing equations, Science China Mathematics, 42 (2012), 13-21.  doi: 10.1360/012011-328.  Google Scholar

[16]

X. Wang, Quasi-periodic solutions for second order differential equation with superlinear asymmetric nonlinearities and nonlinear damping term, Boundary Value Problems, 101 (2015), 1-12.  doi: 10.1186/s13661-015-0370-0.  Google Scholar

[17]

X. Wang, Aubry-Mather sets in semilinear asymmetric Duffing equations, Advances in Difference Equations, 297 (2016), 1-12.  doi: 10.1186/s13662-016-1024-y.  Google Scholar

[18]

Y. Wang, Boundedness of solutions in asymmetric oscillations via the twist theorem, Acta Math. Sinica(English Series), 17 (2001), 313-318.  doi: 10.1007/s101140000043.  Google Scholar

[19]

X. Yang, Boundedness of solutions for sublinear reversible systems, Applied Mathematics and Computation, 158 (2004), 389-396.  doi: 10.1016/j.amc.2003.08.092.  Google Scholar

[20]

X. Yang and K. Lo, Quasi-periodic solutions in nonlinear asymmetric oscillations, Zeitschrift für Analysis und ihre Anwendungen, 26 (2007), 207-220.  doi: 10.4171/ZAA/1319.  Google Scholar

show all references

References:
[1]

A. Capietto and B. Liu, Quasi-periodic solutions of a forced asymmetric oscillator at resonance, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 105-117.  doi: 10.1016/j.na.2003.09.001.  Google Scholar

[2]

A. Capietto, W. Dambrosio and B. Liu, On the boundedness of solutions to a nonlinear singular oscillator, Z. Angew. Math. Phys., 60 (2009), 1007-1034.  doi: 10.1007/s00033-008-8094-y.  Google Scholar

[3]

A. Capietto, W. Dambrosio and X. Wang, Quasi-periodic solutions of a damped reversible oscillator at resonance, Differential and Integral Equations, 22 (2009), 1033-1046.   Google Scholar

[4]

S. Chow and M. Pei, Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems, Nonlinear Analysis: Theory, Methods} & \emph{Applications, 25 (1995), 905-931.  doi: 10.1016/0362-546X(95)00087-C.  Google Scholar

[5]

B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.  doi: 10.1006/jmaa.1998.6219.  Google Scholar

[6]

B. Liu and J. Song, Invariant curved of reversible mappings with small twist, Acta Math. Sinica (English Series), 20 (2004), 15-24.  doi: 10.1007/s10114-004-0316-4.  Google Scholar

[7]

X. Li, Invariant tori for semilinear reversible systems, Nonlinear Analysis, 56 (2004), 133-146.  doi: 10.1016/j.na.2003.09.004.  Google Scholar

[8]

R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.  Google Scholar

[9]

M. Pei, Mather sets for superlinear Duffing equations, Science in China, Ser. A, 36 (1993), 524-537.   Google Scholar

[10]

M. Pei, Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations, 113 (1994), 106-127.  doi: 10.1006/jdeq.1994.1116.  Google Scholar

[11]

D. Qian, Mather sets for sublinear Duffing Equations, Chin. Ann. of Math., Ser. B, 15 (1994), 421-434.   Google Scholar

[12]

J. Si, Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term, Nonlinear Anal., 64 (2006), 1475-1495.  doi: 10.1016/j.na.2005.06.046.  Google Scholar

[13]

X. Wang, Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(English Series), 19 (2003), 765-782.  doi: 10.1007/s10114-003-0249-3.  Google Scholar

[14]

X. Wang, Aubry-Mather sets for semilinear Duffing equations, Acta Mathematica Sinica(Chinese Series), 52 (2009), 605-610.   Google Scholar

[15]

X. Wang, Aubry-Mather sets for sublinear asymmetric Duffing equations, Science China Mathematics, 42 (2012), 13-21.  doi: 10.1360/012011-328.  Google Scholar

[16]

X. Wang, Quasi-periodic solutions for second order differential equation with superlinear asymmetric nonlinearities and nonlinear damping term, Boundary Value Problems, 101 (2015), 1-12.  doi: 10.1186/s13661-015-0370-0.  Google Scholar

[17]

X. Wang, Aubry-Mather sets in semilinear asymmetric Duffing equations, Advances in Difference Equations, 297 (2016), 1-12.  doi: 10.1186/s13662-016-1024-y.  Google Scholar

[18]

Y. Wang, Boundedness of solutions in asymmetric oscillations via the twist theorem, Acta Math. Sinica(English Series), 17 (2001), 313-318.  doi: 10.1007/s101140000043.  Google Scholar

[19]

X. Yang, Boundedness of solutions for sublinear reversible systems, Applied Mathematics and Computation, 158 (2004), 389-396.  doi: 10.1016/j.amc.2003.08.092.  Google Scholar

[20]

X. Yang and K. Lo, Quasi-periodic solutions in nonlinear asymmetric oscillations, Zeitschrift für Analysis und ihre Anwendungen, 26 (2007), 207-220.  doi: 10.4171/ZAA/1319.  Google Scholar

[1]

Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823
[2]

Kaizhi Wang, Lin Wang, Jun Yan. Aubry-Mather theory for contact Hamiltonian systems II. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 555-595. doi: 10.3934/dcds.2021128
[3]

Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
[4]

Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135
[5]

Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[6]

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
[7]

Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[8]

Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
[9]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[10]

Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103
[11]

Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983
[12]

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
[13]

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
[14]

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
[15]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for a class of beam equation system. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 31-53. doi: 10.3934/dcdsb.2019171
[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, 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, 2012, 32 (6) : 2101-2123. doi: 10.3934/dcds.2012.32.2101
[19]

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
[20]

Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (92)
  • HTML views (46)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy