Selfconsistent Transfer Operators for Heterogeneous Coupled Maps

TL;DR

This paper develops a rigorous framework using Self-Consistent Transfer Operators and graphons to analyze the mean-field dynamics of large heterogeneous networks of coupled chaotic maps, linking network structure to emergent behavior.

Contribution

It introduces a novel operator-theoretic approach to study the mean-field limit of heterogeneous coupled maps via graphons, providing insights into equilibrium states and regularity properties.

Findings

Defined a Self-Consistent Transfer Operator (STO) for graphon-based network limits.

Proved existence of attracting fixed points corresponding to equilibrium states.

Linked regularity of graphons to stability and properties of fixed points.

Abstract

We investigate the dynamics of large heterogeneous network dynamical systems composed of nonlocally coupled chaotic maps. We show that the mean-field limit of such systems is governed by a suitably defined Self-Consistent Transfer Operator (STO) acting on graphons, describing the infinite-size limits of dense graphs, thereby allowing for a rigorous analysis of the system as the network size tends to infinity. We construct appropriate functional spaces on which the STO has an attracting fixed point, which corresponds to the equilibrium state for the mean-field limit, and we draw a connection between the regularity properties of the graphons and the regularity of the fixed points. This work combines operator theory and graph limits tools to offer a framework for understanding emergent behavior in complex networks.