Topics in Applied Stochastic Dynamics.

Abstract

Randomness in natural systems come from various sources, for example from the discrete nature of the underlying dynamical process when viewed on a small scale. In this thesis we study the effect of stochasticity on the dynamics in three applications, each with different sources and effects of randomness. In the first application we study the Hodgkin-Huxley model of the neuron with a random ion channel mechanism via numerical simulation. Randomness affects the nonlinear mechanism of a neuron’s firing behavior by spike induction as well as by spike suppression. The sensitivity to different types of channel noise is explored and robustness of the dynamical properties is studied using two distinct stochastic models. In the second application we compare and contrast the effectiveness of mixing of a passive scalar by stirring using different notions of mixing efficiency. We explore the non-commutativity of the limits of large Peclet numbers and large spatial scale separation between the flow and sources and sinks, and propose and examine a conceptual approach that captures some compat- ible features of the different models and measures of mixing. In the last application we design a stochastic dynamical system that mimics the properties of so-called ho- mogeneous Rayleigh-Benard convection and show that arbitrary small noise changes the dynamical properties of the model. The system’s properties are further exam- ined using the first exit time problem. The three applications show that randomness of small magnitude may play important and counterintuitive roles in determinig a system’s properties.