One-dimensional random attractor and rotation number of the stochastic damped sine-Gordon equation

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:248

Article

Shen, Zhongwei¹  Zhou, Shengfan¹  Shen, Wenxian² 

[1] Shanghai Normal Univ, Dept Appl Math, Shanghai 200234, Peoples R China

[2] Auburn Univ, Dept Math & Stat, Auburn, AL 36849 USA

关键词: Stochastic damped sine-Gordon equation;    Random horizontal curve;    One-dimensional random attractor;    Rotation number;    Frequency locking;   

DOI  :  10.1016/j.jde.2009.10.007

来源: Elsevier

【 摘 要 】

This paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine-Gordon equation with homogeneous Neumann boundary condition. It is shown that for any positive damping and diffusion coefficients, the equation possesses a random attractor, and when the damping and diffusion coefficients are sufficiently large, the random attractor is a one-dimensional random horizontal curve regardless of the strength of noise. Hence its dynamics is not chaotic. It is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large, which implies that the solutions tend to oscillate with the same frequency eventually and the so-called frequency locking is successful. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free