Ph.D. Course Details

The course is about a topic which is currently fastly developing in the research literature. The course is a doctorate one, but it can also be followed with profit by master students and it will include a seminar given by Bastien Fernandez (Université Paris Cité).

Below we add a description of the content and a list of prerequisites needed. Exams can be organized online or in presence, according to the students needs.

Students interested in following the course remotely can email Prof. Stefano Galatolo asking to be added to the course Teams group.

A complex system is typically a system in which many subsystems interact, and whose evolution is substantially determined by this interaction. The typical characteristic of these systems is the emergence of global behaviors of the system that are due to the interaction among the components and that are not explainable/predictable from the characteristics of the individual subsystems (emergent properties). Coupled maps exhibit a variety of behaviors reminiscent of those of complex systems: from regular ones such as synchronization, to “random” and uncorrelated evolution of certain components. In this course, we will present some topics on the study of coupled maps with mean-field interactions and study, within the framework of ergodic theory, the mechanisms that explain certain behaviors of complex systems. In the first part, we will introduce some general concepts of ergodic theory, with particular attention to the study of transfer operators of hyperbolic systems. In the second part, these results will be extended to the study of the statistical properties of coupled maps. Rigorous results on these systems have been obtained in the thermodynamic limit when the number of coupled maps tends to infinity. In this limit, the dynamical system is of infinite dimension, and its evolution is prescribed by a self-consistent transfer operator. We will study the main techniques available to study these objects and discuss how to prove the existence of fixed points (corresponding to equilibria in the thermodynamic limit), their stability (i.e., whether nearby states tend to converge to the fixed point or not), and their stability under perturbation (i.e., how the fixed point changes when the equations defining the system change).

Contents

• Recall of ergodic theory
• Transfer operators, invariant measures, convergence to equilibrium, functional-analytic approach
• Statistical stability, Linear Response to perturbations
• Interacting systems in mean field, self-consistent operators
• Coupled maps systems, examples
• Emergent properties of coupled systems, synchronization phenomena
• Statistical properties of coupled systems: invariant measures, convergence to equilibrium
• Statistical stability and linear response to perturbations

Prerequisites

• Basic notions on the topology of metric spaces
• Basic notions of measure theory and functional anaysis (measures, Lebesgue integral and its main properties, L^p spaces, basic notions of Sobolev spaces, bounded operators and the basics of their spectral properties)
• Basic knowledge of what is a differential equation and a dynamical system

Available teaching resources

• Lecture notes about a large part of the course will be provided
• The remaining material will be provided giving references to openly accessible literature