Nonlinear Dynamics with Noise

The following project proposal is intended for third-year students taking the mathematical sciences final-year undergraduate project module (3H level) in the coming academic year (2023/2024).

Simple nonlinear systems can display a wide array of interesting dynamical behaviors and serve as invaluable tools for understanding the mechanisms behind phenomena in complex real-world systems. In many applications, our system of interest is “open” in the sense that it is subject to external forcing of some kind; this naturally leads to the study of stochastic differential equations of the form:

$\centering dx(t) = f(x(t))\,dt + \sigma(x(t))\,dB(t),$

where the function $f$ represents the deterministic nonlinear response and $B(t)$ is a stochastic process capturing the impact of noise on the system.

The premise of this project is to investigate noise-induced phenomena in simple dynamical systems arising in different real-world applications. For example, Figure 1A below shows some trajectories of a simple ecological model with two alternative stable states, forest and grassland, in which we observe noise-induced switching between these two stable states (blue trajectory). Figure 1B shows trajectories of the Fitzhugh-Nagumo model of neuronal dynamics in the phase space. In this case, we observe noise-induced oscillations (blue trajectory) in a parameter regime which would show no oscillations with zero noise (red trajectory ends at a fixed point).

A: Simulations of an ecological model of forest-grass interactions; deterministic trajectories in red and stochastic in blue. B: Trajectories of the Fitzhugh-Nagumo model with sustained oscillations when subject to noise (blue) and tending to a stable fixed point without noise (red).

Students will begin by learning how to simulate solutions to stochastic differential equations (in MATLAB, Python or a similar scripting language). They will then numerically study some paradigmatic examples of noise-induced dynamics in well-known models. Depending on individual interests, students could decide to analyze more complex examples (e.g. spatial or network models), build their own model system for investigation, or delve into more analytic aspects of stochastic dynamics (e.g. Fokker-Planck equations, rigorous results in the small noise limit, etc.).

Prerequisites

Students should have a solid foundation in dynamical systems (familiarity with basic bifurcations) and probability theory (random variables and standard distributions). Prior coding experience and an interest in mathematical modeling in the applied sciences would both be useful.

Resources

Nonlinear Dynamics and Chaos by Steven Strogatz is a standard introductory text on nonlinear dynamics.

Stochastic Modeling of Reaction-Diffusion Processes by Erban and Chapman contains all of the theory needed for the simulation aspect of this project and some good examples of the types of phenomena that could be investigated.

Lindner, B., Garcıa-Ojalvo, J., Neiman, A. and Schimansky-Geier, L. (2004). Effects of noise in excitable systems. Physics reports, 392(6), pp.321-424. [review of noise-induced phenomena in neuroscience]