# American Institute of Mathematical Sciences

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

 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
 $p$
.
Citation: Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete and 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. [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. [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. [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. [5] B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.  doi: 10.1006/jmaa.1998.6219. [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. [7] X. Li, Invariant tori for semilinear reversible systems, Nonlinear Analysis, 56 (2004), 133-146.  doi: 10.1016/j.na.2003.09.004. [8] R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325. [9] M. Pei, Mather sets for superlinear Duffing equations, Science in China, Ser. A, 36 (1993), 524-537. [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. [11] D. Qian, Mather sets for sublinear Duffing Equations, Chin. Ann. of Math., Ser. B, 15 (1994), 421-434. [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. [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. [14] X. Wang, Aubry-Mather sets for semilinear Duffing equations, Acta Mathematica Sinica(Chinese Series), 52 (2009), 605-610. [15] X. Wang, Aubry-Mather sets for sublinear asymmetric Duffing equations, Science China Mathematics, 42 (2012), 13-21.  doi: 10.1360/012011-328. [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. [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. [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. [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. [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.

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##### 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. [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. [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. [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. [5] B. Liu, Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.  doi: 10.1006/jmaa.1998.6219. [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. [7] X. Li, Invariant tori for semilinear reversible systems, Nonlinear Analysis, 56 (2004), 133-146.  doi: 10.1016/j.na.2003.09.004. [8] R. Ortega, Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325. [9] M. Pei, Mather sets for superlinear Duffing equations, Science in China, Ser. A, 36 (1993), 524-537. [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. [11] D. Qian, Mather sets for sublinear Duffing Equations, Chin. Ann. of Math., Ser. B, 15 (1994), 421-434. [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. [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. [14] X. Wang, Aubry-Mather sets for semilinear Duffing equations, Acta Mathematica Sinica(Chinese Series), 52 (2009), 605-610. [15] X. Wang, Aubry-Mather sets for sublinear asymmetric Duffing equations, Science China Mathematics, 42 (2012), 13-21.  doi: 10.1360/012011-328. [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. [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. [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. [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. [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.
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