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The aim of this paper is to rigorously study the dynamics of Heterogeneously Coupled Maps (HCM). Such systems are determined by a network with heterogeneous degrees. Some nodes, called hubs, are very well connected while most nodes interact with few others. The local dynamics on each node is chaotic, coupled with other nodes according to the network structure. Such high-dimensional systems are hard to understand in full, nevertheless we are able to describe the system over exponentially large time scales. In particular, we show that the dynamics of hub nodes can be very well approximated by a low-dimensional system. This allows us to establish the emergence of macroscopic behaviour such as coherence of dynamics among hubs of the same connectivity layer (i.e. with the same number of connections), and chaotic behaviour of the poorly connected nodes. The HCM we study provide a paradigm to explain why and how the dynamics of the network can change across layers.
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Tiago Pereira, Sebastian van Strien, Matteo Tanzi, Heterogeneously coupled maps: hub dynamics and emergence across connectivity layers. J. Eur. Math. Soc. 22 (2020), no. 7, pp. 2183–2252DOI 10.4171/JEMS/963