abstract: Heterogeneously Coupled Maps 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 hyperbolic and coupled with other nodes according to the network structure. For such high-dimensional systems there is a regime of the interaction strength such that the poorly connected systems feel a small perturbation, while for the hub nodes the sum of all interactions can be large. In particular, global hyperbolicity might be lost.
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 hyperbolic 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.