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

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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 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.

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.
[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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