Stochastic Center Manifold Analysis in Scalar Nonlinear Systems Involving Distributed Delays and Additive Noise

A. Hutt, J. Lefebvre

2016, v.22, Issue 3, 555-572

ABSTRACT

The work reviews and extends a recent analysis center manifold technique that allows to describe stochastic bifurcations in delayed systems induced by additive noise. In contrast to previous studies, the present study considers distributed delays while focussing on stochastic Hopf bifurcations. We motivate the analysis by a spatially extended neural field model involving distributed delays and observe a stochastic bifurcation induced by the additive noise.
This study reviews and extends a recent center manifold analysis technique developed to characterize stochastic bifurcations in delayed systems induced by additive noise. Motivated by the dynamics of spatially extended neural field models with finite propagation velocity, we revealed and fully characterized codimension 1 stochastic bifurcations induced by additive white noise. In contrast to previous studies, we here extended our analysis to the case of distributed delays while applying our results to the stochastic Hopf bifurcation. Taken together, our results provide further insight on the conjugate role of noise and delays in the genesis non-linear phenomena.

Keywords: stochastic bifurcation, neural fields, delay equations, bifurcation theory

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