On the pitchfork bifurcation for the Chafee-Infante equation with additive noise. (English) Zbl 1531.37044
Summary: We investigate pitchfork bifurcations for a stochastic reaction diffusion equation perturbed by an infinite-dimensional Wiener process. It is well-known that the random attractor is a singleton, independently of the value of the bifurcation parameter; this phenomenon is often referred to as the “destruction” of the bifurcation by the noise. Analogous to the results of M. Callaway et al. [Ann. Inst. Henri Poincaré, Probab. Stat. 53, No. 4, 1548–1574 (2017; Zbl 1383.37037)] for a 1D stochastic ODE, we show that some remnant of the bifurcation persists for this SPDE model in the form of a positive finite-time Lyapunov exponent. Additionally, we prove finite-time expansion of volume with increasing dimension as the bifurcation parameter crosses further eigenvalues of the Laplacian.
MSC:
| 37H20 | Bifurcation theory for random and stochastic dynamical systems |
| 37L55 | Infinite-dimensional random dynamical systems; stochastic equations |
| 37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60H50 | Regularization by noise |
Keywords:
stochastic partial differential equations; singleton attractors; Lyapunov exponents; stochastic bifurcationsCitations:
Zbl 1383.37037References:
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