1. On the pitchfork bifurcation for the Chafee–Infante equation with additive noise.
- Author
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Blumenthal, Alex, Engel, Maximilian, and Neamţu, Alexandra
- Subjects
- *
REACTION-diffusion equations , *STOCHASTIC partial differential equations , *WIENER processes , *HEAT equation , *LYAPUNOV exponents , *EQUATIONS , *NOISE - 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 Prob 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. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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