A stochastic pitchfork bifurcation in a reaction-diffusion equation
UNSPECIFIED. (2001) A stochastic pitchfork bifurcation in a reaction-diffusion equation. PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 457 (2013). pp. 2041-2061. ISSN 1364-5021Full text not available from this repository.
We study in some detail the structure of the random attractor for the Chafee-Infante reaction-diffusion equation perturbed by a multiplicative white noise,
du = (Deltau + betau - u(3)) dt + sigmau circle dW(t), x is an element of D subset of R-m.
First we prove., for m less than or equal to 5, a lower bound on the dimension of the random attractor, which is of the same order in beta as the upper bound we derived in an earlier paper. and is the same as that obtained in the deterministic case. Then we show, for m = 1. that as beta passes through lambda (1) (the first eigenvalue of the negative Laplacian) from below, the system undergoes a stochastic bifurcation of pitchfork type. We believe that this is the first example of such a stochastic bifurcation in an infinite-dimensional setting. Central to our approach is the existence of a random unstable manifold.
|Item Type:||Journal Article|
|Journal or Publication Title:||PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES|
|Publisher:||ROYAL SOC LONDON|
|Official Date:||8 September 2001|
|Number of Pages:||21|
|Page Range:||pp. 2041-2061|
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