A stochastic pitchfork bifurcation in a reaction-diffusion equation

By Tomás Caraballo, José A. Langa, James C. Robinson. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 2001

106 citations recorded by the source corpus.

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=( Δu+βu- u 3 ) dt+σuod W t ,x∈D⊂ R m First we prove, for m ⩽ 5, a lower bound on the dimension of the random attractor, which is of the same order in β 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 β passes through λ1 (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.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/f100d39e52c71add8fbd03b7a0917b30d3815a50

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

In Computer Networks and Communications

In Control and Systems Engineering

Citations in the corpus

Asymptotic behavior of stochastic wave equations with critical exponents on $\mathbb{R}^{3}$ (2008, unknown): “[1, 8 , 9, 10, 11, 17] and the references therein.”
INVARIANT MANIFOLDS FOR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS (2003, unknown): “In Caraballo et al. [ 22 ] an invariant manifold for a stochastic reaction diusion equation of pitchfork type has been considered.”
Smooth Stable and Unstable Manifolds for Stochastic Evolutionary Equations (2004, methodology): “Similar transformations were used in Caraballo, Langa and Robinson [ 5 ], and Schmalfus [19].”
Invariant Manifolds for Random and Stochastic Partial Differential Equations (2009, background): “Stable, unstable, center, and inertial manifolds, as special random invariant structures, have been considered in the investigation of stochastic partial differential equations or stochastic evolutionary equations in infinite dimensional spaces [8, 12, 13, 19, 5, 15, 11, 27].”
Random attractors for stochastic reaction–diffusion equations on unbounded domains (2009, background): “, [10,14,15,17] and the references therein, but little is known for unbounded domains.”
Climate dynamics and fluid mechanics: Natural variability and related uncertainties (2008, background): “Thus RDS concepts and tools are not restricted to finite-dimensional systems [104, 105, 106].”
Existence, Stability and Bifurcation of Random Complete and Periodic Solutions of Stochastic Parabolic Equations (2013, background): “5) have been investigated in [13, 14, 34] for α = 0 and in [12] for constant γ.”