The mathematical study of coupled dynamical systems, i.e. systems with multiple components that interact with
one another, started with the formulation of Newton’s universal law of gravitation when astronomers and mathematicians
addressed the question on the stability of the Solar System. Since then, it has become clear that even weak
interactions between different agents can have drastic consequences on their evolution and make their dynamics very
hard to predict. Nowadays, models of networks of coupled dynamical systems appear in many areas of science:
physics, chemistry, biology and also engineering and human sciences. Examples range from solid state physics to
astrophysics, reaction-diffusion equations, transportation systems, the Internet, social networks, opinion models, etc.
This vast spectrum of applications has stimulated a broad interdisciplinary endeavour to predict and control the behaviour
of so-called ”complex systems”, i.e. large ensembles of units coupled through an intricate interaction web.
The main goal of this project is to study networks of coupled dynamical systems, and how the interplay between
structure of the interactions and local dynamics shapes their evolution. The systems under study are inspired by
paradigmatic examples in biology, and relate to questions of primary importance in this field. For example: how does
the dynamics of coupled neurones shape the functionality of the brain? How does it give rise to stable spiking patterns
such as gamma-band oscillations, and, at the same time, undergo transitions from healthy to pathological states?
A rigorous mathematical framework to satisfactorily explain these mechanisms is currently
not available.
So far, most studies on coupled systems have been limited to numerical methods due to the considerable difficulties
encountered in formulating general theoretical descriptions. Instead, this project aims to develop rigorous
mathematical approaches, accompanied with computational methods when necessary, to describe the emergence of
global behaviour in complex heterogeneous systems, underlining its dependence on the microscopic features. At the
interface between pure and applied mathematics, the strategy is to rely on results from abstract ergodic theory, especially
those recently obtained for non-uniformly hyperbolic systems, to address concrete archetypes exhibiting
real-world features.
