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Topological stability criteria for networking dynamical systems with Hermitian Jacobian

Published online by Cambridge University Press:  02 November 2016

A. L. DO ,
S. BOCCALETTI ,
J. EPPERLEIN ,
S. SIEGMUND  and
T. GROSS
A. L. DO
Affiliation:
Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany email: ly@pks.mpg.de
S. BOCCALETTI
Affiliation:
Institute of Complex Systems of the CNR, Florence, Italy email: stefano.boccaletti@gmail.com The Italian Embassy in Israel, Tel Aviv, Israel email: jeremias.epperlein@tu-dresden.de
J. EPPERLEIN
Affiliation:
Dresden University of Technology, Institute of Analysis, Dresden, Germany email: stefan.siegmund@tu-dresden.de
S. SIEGMUND
Affiliation:
Dresden University of Technology, Institute of Analysis, Dresden, Germany email: stefan.siegmund@tu-dresden.de
T. GROSS
Affiliation:
University of Bristol, Merchant Venturers School of Engineering, Bristol, UK email: thilo2gross@gmail.com
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Abstract

The central theme of complex systems research is to understand the emergent macroscopic properties of a system from the interplay of its microscopic constituents. The emergence of macroscopic properties is often intimately related to the structure of the microscopic interactions. Here, we present an analytical approach for deriving necessary conditions that an interaction network has to obey in order to support a given type of macroscopic behaviour. The approach is based on a graphical notation, which allows rewriting Jacobi's signature criterion in an interpretable form and which can be applied to many systems of symmetrically coupled units. The derived conditions pertain to structures on all scales, ranging from individual nodes to the interaction network as a whole. For the purpose of illustration, we consider the example of synchronization, specifically the (heterogeneous) Kuramoto model and an adaptive variant. The results complete and extend the previous analysis of Do et al. (2012Phys. Rev. Lett.108, 194102).

Keywords

stability analysiscomplex networksdefiniteness of matrices
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Special Issue 6: Network Analysis and Modelling , December 2016 , pp. 888 - 903
Copyright
Copyright © Cambridge University Press 2016 

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