Strongly Coupled Maps: Thermodynamic Limit and Rates of Convergence

A complex system consists of a large number of individual components that interact with one another and whose global behaviour emerging from these interactions is not apparent from the laws that govern the components in isolation. Climate change, spread of epidemics, criminal networks, neurological disorders and onset of diseases are all examples of complex systems with a potentially catastrophic impact on the livelihoods of human communities. Novel interdisciplinary approaches are on demand to understand how interactions lead to the observed global behaviour. Achieving this can help to prevent disasters, or inform strategies to minimise impact with limited resources. The potential for impact has recently put complexity science under the radar of governments and international organisations like the UN and UNICEF. Models of complex systems are often prescribed by a network where each node represents a component and the links prescribe the coupling structure of the interactions among the units. One of the main questions in the field is to find how the network structure and type of interactions shape the time evolution of a complex system. This is notoriously challenging to determine due to the large number of components and the intricacy of their interactions. This proposal addresses this question from the rigorous mathematical perspective and studies discrete time models of complex systems given by strongly coupled components. It considers systems presenting chaos — a staple of complexity — where, although deterministic, each component presents erratic behaviour due to the nonlinearity in its evolution laws. This proposal focuses on the challenges faced when dealing with large coupling strength between the components. The proposed investigation has two complementary aims: (1) to determine sufficient conditions for the emergence of equilibrium states in the thermodynamic limit where the number of units composing the system is infinite; (2) to quantify the rates of convergence to the thermodynamic limit and obtain control over the evolution of finite (but large) dimensional systems. This will allow us to answer questions like: What are the stable equilibrium states we expect to emerge in a complex system? How does their stability change changing the coupling strength? What are the correlations among coordinates in terms of the type and strength of the interactions? The aims will be achieved by combining smooth ergodic theory (evolution of measure disintegrations on non-invariant foliations), probability (large deviations and concentration of measure), and analysis (study of the contraction properties of nonlinear operators).