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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

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