TY - JOUR
T1 - Topological stability criteria for networking dynamical systems with Hermitian Jacobian
AU - Do, A. L.
AU - Boccaletti, S.
AU - Epperlein, J.
AU - Siegmund, S.
AU - Gross, T.
PY - 2016/12/1
Y1 - 2016/12/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84994210549&partnerID=8YFLogxK
U2 - 10.1017/S0956792516000425
DO - 10.1017/S0956792516000425
M3 - Article (Academic Journal)
AN - SCOPUS:84994210549
SN - 0956-7925
VL - 27
SP - 888
EP - 903
JO - European Journal of Applied Mathematics
JF - European Journal of Applied Mathematics
IS - 6
ER -