ErgThComplexSys · Ergodic theory for complex systems: a rigorous study of dynamics on heterogeneous networks

Ergodic theory for complex systems: a rigorous study of dynamics on heterogeneous networks

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.

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