Stochastic simulation, analysis, and inference of non-linear dynamical systems

This project will develop a novel framework for studying the dynamics of systems presenting oscillations and multi-stabilities. This type of dynamics is abundant in many fields and especially in molecular biology, epidemiology, ecology, sociology. For instance, the developed methods will apply to gene expression oscillations for biological time-keeping and cell-to-cell communication, multi-stabilities in cell development, epidemic oscillations driven by public awareness, ecological oscillations driven by species competitions, and many other settings. To build this framework, we will use powerful results from the theory of dynamical systems to decompose large, non-linear dynamical systems. This decomposition will allow us to break down systems into simpler components. For the non-linear components presenting oscillations or multi-stabilities, we will now be able to study their dynamics in detail and develop methods for controlling their non-linear variation, while for the linear components use standard models described by stochastic differential equations. The combination of the two components will provide fast, and long-time accurate models for a much wider range of problems than ever before. We will then be able to use these models for fast, long-time accurate simulation and parameter estimation algorithms.

The ideal candidate for this project will be interested in stochastic processes and dynamical systems. Background in stochastic processes (e.g. Markov processes, stochastic differential equations) or non-linear dynamical systems will be beneficial, but candidates with strong background in other mathematical subjects will be considered.