Delayed bifurcation onset of Turing instability and the effect of noise

Abstract

In this thesis we study the effect of noise on delayed bifurcations in PDE's. Two particular problems with slowly varying parameter and noise are considered: a single PDE and a modified version of Klausmeier model. We first study the combined effect of spatio-temporal noise and slowly varying parameter on the onset of the Turing bifurcation. The noise helps to trigger the spatially in-homogeneous instability, and we compute the delay in this instability. We review and extend the results of [1], to more general situations, and also analyse the full distribution of the blowup time. We also analyse the case where spatio-temporal noise is replaced by purely spatial noise and we obtain the asymptotic density distribution of blow up time in the case where the domain is sufficiently small. Full numerical simulations are used to validate our analytical results.