Attractors in random dynamical systems with bounded noise: persistence, resilience and bifurcations

Dynamical systems with noise (random dynamical systems) arise naturally in the mathematical modelling of a large range of natural and societal phenomena, from Brownian motion, to climate and the financial markets. However, the mathematical tools available to study random dynamical systems are limited.

In the absence of noise, bifurcation theory provides many insights into the changes in behaviour of a dynamical system. Numerical bifurcation analysis tools have become popular in many areas of science and engineering, including industry. However, the established bifurcation theory does not generalise in any obvious way to systems with noise.

To overcome this challenge, this proposal builds on recent mathematical advances in random dynamical systems with bounded noise. The key insight is that the boundary of an attractor is composed of (parts of) invariant sets of a deterministic finite-dimensional map, the so-called boundary map, which is in principle amenable to established analysis techniques. This elegant mathematical fact will be developed further alongside its application to physically-motivated pilot studies. The ultimate aim of this proposal is to make bifurcation analysis tools for random dynamical systems with bounded noise as common as those for deterministic dynamical systems, and create opportunities for their exploitation across science, engineering, and beyond.