Dynamics Of The Stochastic Damped Sine-Gordon Equation

This paper is devoted to the study of the asymptotic behavior of the stochastic damped sine-Gordon equation with homogeneous Neumann boundary condition. More precisely, the existence of random attractor and its structure.It is shown first that for any positive damping and diffusion coefficients, the random dynam-ical system determined by the equation possesses a unique random attractor, which attracts all pseudo-tempered random sets in the phase space, in a general sense because of the uncontrolled space average of the solutions. More precisely, the equation has a random attractor when its corresponding equation on torus has a random attractor.Moreover, by studying the invariant manifold corresponding to the zero eigenvalue and its stable foliations, we show that 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.Finally, it is also shown by using related theory of random probability measure and ergodic theory 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.