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Stochastic D-bifurcation for a damped sine-Gordon equation with noise
1.T. Visser, Modelling and Analysis of Long Josephson Junctions (Twente University Press, 2002).
4.D. Blömker, Amplitude Equations for Stochastic Partial Differential Equations (World Scientific, 2007).
6.H. Crauel, Stochastic Dynamics (Springer, 1999).
8.L. Arnold, Random Dynamical Systems (Springer-Verlag, 1998).
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We investigate the stochastic
bifurcation of a damped sine-Gordon equation with Dirichlet boundary conditions under the influence of multiplicative Gaussian white noise. Introducing a slow time scale, we derive the amplitude equations near the trivial solution by multiscale analysis. And the stationary probability density functions are formulated analytically using the stochastic averaging of energy envelope. The numerical calculations show that the system undergoes a stochastic D-bifurcation of energy envelope from a delta measure to new stationary measures when the control parameter crosses a critical point.
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