Thursday 10 Nov 2016: Dynamics seminar: Mean Field Reduction for the Hub Dynamics of Expanding Maps Coupled in Heterogeneous Networks
Matteo Tanzi - Imperial College London
Streatham Court C 14:30-16:30
We study expanding dynamical systems coupled via a diffusive coupling where the presence of interaction between two systems is prescribed by the edges of an heterogeneous network. In this setting, most systems interact with few others, while those corresponding to the highly connected nodes (hubs) interact with many. We study the regime where the coupling is a small perturbation for the former systems, but a large one for the latter so that global system is not hyperbolic. Using transfer operator techniques, we show that, under given hypotheses, in the limit of finite, but very large network, the effect of the coupling on the hub nodes can be well approximated, for long time and a large set of initial conditions, as the mean contribution coming from the other nodes. This implies a phase transition for the observed behaviour of the hub nodes when varying the coupling strength and forecast regimes of synchronisation and uncorrelation of the nodes.