Quasiperiodic solutions of semilinear Liénard equations

Previous Article
A one-parameter family of analytic Markov maps with an intermittency transition
DCDS Home
This Issue
Next Article
On generalized Benjamin type equations

January 2005, 12(1): 137-160. doi: 10.3934/dcds.2005.12.137

Quasiperiodic solutions of semilinear Liénard equations

1.	LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

Received June 2003 Revised May 2004 Published December 2004

Abstract
Related Papers

We deal with the existence of quasi-periodic solutions in classical sense and in the generalized sense, i.e., the existence of invariant tori and Aubry-Mather sets for some semilinear differential equations

$ x'' + F_x(x,t)x'+ a^2x + \phi(x) + e(x,t) = 0, $

where $F$ and $e$ are smooth and $2\pi$-periodic in $t$ and $a>0$ is a constant. As a consequence, we also get the boundedness of all the solutions.

Keywords: quasi-periodic solutions, reversible systems, Aubry-Mather set, Liénard equations., boundedness of solutions.

Mathematics Subject Classification: Primary 34C15, 58F2.

Citation: Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137

[1]	Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703
[2]	Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393
[3]	Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563
[4]	Claudia Valls. On the quasi-periodic solutions of generalized Kaup systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 467-482. doi: 10.3934/dcds.2015.35.467
[5]	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
[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]	Xiaoping Yuan. Quasi-periodic solutions of nonlinear wave equations with a prescribed potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 615-634. doi: 10.3934/dcds.2006.16.615
[8]	Ugo Bessi. Viscous Aubry-Mather theory and the Vlasov equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 379-420. doi: 10.3934/dcds.2014.34.379
[9]	Hans Koch, Rafael De La Llave, Charles Radin. Aubry-Mather theory for functions on lattices. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 135-151. doi: 10.3934/dcds.1997.3.135
[10]	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
[11]	Francesca Alessio, Piero Montecchiari, Andrea Sfecci. Saddle solutions for a class of systems of periodic and reversible semilinear elliptic equations. Networks & Heterogeneous Media, 2019, 14 (3) : 567-587. doi: 10.3934/nhm.2019022
[12]	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
[13]	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
[14]	Fabio Camilli, Annalisa Cesaroni. A note on singular perturbation problems via Aubry-Mather theory. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 807-819. doi: 10.3934/dcds.2007.17.807
[15]	Artur O. Lopes, Rafael O. Ruggiero. Large deviations and Aubry-Mather measures supported in nonhyperbolic closed geodesics. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1155-1174. doi: 10.3934/dcds.2011.29.1155
[16]	Bassam Fayad. Discrete and continuous spectra on laminations over Aubry-Mather sets. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 823-834. doi: 10.3934/dcds.2008.21.823
[17]	Diogo A. Gomes. Viscosity solution methods and the discrete Aubry-Mather problem. Discrete & Continuous Dynamical Systems - A, 2005, 13 (1) : 103-116. doi: 10.3934/dcds.2005.13.103
[18]	Siniša Slijepčević. The Aubry-Mather theorem for driven generalized elastic chains. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2983-3011. doi: 10.3934/dcds.2014.34.2983
[19]	Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
[20]	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

2018 Impact Factor: 1.143

American Institute of Mathematical Sciences