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 & 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. CapiettoW. 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. CapiettoW. 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. CapiettoW. 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. CapiettoW. 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

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