Heterogeneously coupled maps. From high to low dimensional systems through ergodic theory
File(s)
Author(s)
Tanzi, Matteo
Type
Thesis
Abstract
In this thesis we study ergodic theoretical properties of high-dimensional systems coupled on graphs. The local dynamics at each node is hyperbolic and coupled with other nodes according to the edges of the graph. We focus our attention on the case of graphs with heterogeneous degrees meaning that most of the nodes make a small number of interactions, while a few hub nodes have very high degree. For such high-dimensional systems there is a regime of the interaction strength for which the coupling is small for poorly connected systems, and large for the hub nodes. In particular, global hyperbolicity might be lost. We show that, under certain hypotheses, the dynamics of hub nodes can be very well approximated by a low-dimensional system for exponentially long time in the size of the network and that the system exhibit hyperbolic behaviour in this time window. Even if this describes only a long transient, we argue that this is the behaviour that one expects to observe in experiments. Such a description 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). The HCM we study provide a new paradigm to explain why and how the dynamics of a network dynamical system can change across layers.
Version
Open Access
Date Issued
2017-06
Date Awarded
2017-08
Copyright Statement
Attribution NoDerivatives 4.0 International Licence (CC BY-ND)
Advisor
van Strien, Sebastian
Pereira, Tiago
Sponsor
Imperial College London; European Union
Grant Number
ERC AdG grant no 339523 RGDD
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)