Bifurcations of random dynamical systems with bounded noise

The development of dynamical systems theory has been one of the scientific revolutions of the 20th century, and many insights from this field are now at the heart of deep and abstract mathematics as well as computational methodologies in many branches of science and engineering. During the last decades, the importance of considering the presence of noise and uncertainty has become evident in real-world applications, including the life sciences and artificial intelligence, but a corresponding random dynamical systems theory is still only in its very early stages of development.

Bifurcation theory addresses the fundamental question of how the dynamics of a system changes under variation of parameters. While bifurcation theory is a cornerstone of the deterministic theory of dynamical systems, a corresponding theory in the presence of noise is still in its infancy, despite its relevance to many topical applications. This is due to the fact that the deterministic theory does not generalise in any obvious way to the random setting.

The compound behaviour of a random dynamical system with bounded noise - describing the collection of trajectories with all possible noise realisations - admits a description at the topological level as a deterministic set-valued dynamical system. Attractors of this set-valued system correspond to attractors of the associated random system. However, set-valued dynamical systems are notoriously difficult to analyse, since they are defined on the set of all compact subsets, which is not a Banach space. This prevents the use of the implicit function theorem and other powerful tools fundamental for bifurcation theory (and, in general, qualitative dynamical systems theory) and is a well-known obstacle for theoretical and numerical methods alike. As a consequence, the tracking and bifurcation analysis of attractors of set-valued dynamical systems is a notoriously hard problem.

We propose to use a novel insight to resolve the challenge of bifurcation analysis in set-valued dynamics, namely that boundaries of attractors of a dynamical system with bounded noise (and its associated set-valued dynamical system) admit a representation in terms of projected invariant manifolds of a related finite-dimensional dynamical system, defined on the unit tangent bundle over the state space. This boundary map opens up the possibility of studying bifurcations of attractors of systems with bounded noise, thus enabling the use of existing techniques of local and global bifurcation theory in the latter setting.