We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Specifically, each additional term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncated series grows geometrically with the second largest eigenvalue of the normalized adjacency matrix, analogously to the rate of convergence to the stationary distribution of a random walk. Last, we derive a local approximation for the synchronized state by truncating the spatial series, at the first neighborhood term, to illustrate the practical advantages of our approach.
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June 2021
Research Article|
June 07 2021
Geometric unfolding of synchronization dynamics on networks
Lluís Arola-Fernández
;
Lluís Arola-Fernández
1
Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili
, 43007 Tarragona, Spain
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Per Sebastian Skardal;
Per Sebastian Skardal
2
Department of Mathematics, Trinity College
, Hartford, Connecticut 06106, USA
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Alex Arenas
Alex Arenas
a)
1
Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili
, 43007 Tarragona, Spain
a)Author to whom correspondence should be addressed: alexandre.arenas@urv.cat
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a)Author to whom correspondence should be addressed: alexandre.arenas@urv.cat
Chaos 31, 061105 (2021)
Article history
Received:
April 12 2021
Accepted:
May 17 2021
Citation
Lluís Arola-Fernández, Per Sebastian Skardal, Alex Arenas; Geometric unfolding of synchronization dynamics on networks. Chaos 1 June 2021; 31 (6): 061105. https://doi.org/10.1063/5.0053837
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