On the pitchfork bifurcation for the Chafee-Infante equation with additive noise
Probability
2023-10-26 v3 Dynamical Systems
Abstract
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 [Callaway et al., AIHP Probab. Stat., 53:1548-1574, 2017] 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.
Keywords
Cite
@article{arxiv.2108.11073,
title = {On the pitchfork bifurcation for the Chafee-Infante equation with additive noise},
author = {Alex Blumenthal and Maximilian Engel and Alexandra Neamtu},
journal= {arXiv preprint arXiv:2108.11073},
year = {2023}
}