Files and links (1)
Published, Version of Record (VoR)Open Access (License Unspecified), Open
Journal article
THE MEAN FIELD ANALYSIS OF THE KURAMOTO MODEL ON GRAPHS II. ASYMPTOTIC STABILITY OF THE INCOHERENT STATE, CENTER MANIFOLD REDUCTION, AND BIFURCATIONS
Discrete and continuous dynamical systems. Series A, v 39(7), pp 3897-3921
01 Jul 2019
Abstract
In our previous work [3], we initiated a mathematical investigation of the onset of synchronization in the Kuramoto model (KM) of coupled phase oscillators on convergent graph sequences. There, we derived and rigorously justified the mean field limit for the KM on graphs. Using linear stability analysis, we identified the critical values of the coupling strength, at which the incoherent state looses stability, thus, determining the onset of synchronization in this model.
In the present paper, we study the corresponding bifurcations. Specifically, we show that similar to the original KM with all-to-all coupling, the onset of synchronization in the KM on graphs is realized via a pitchfork bifurcation. The formula for the stable branch of the bifurcating equilibria involves the principal eigenvalue and the corresponding eigenfunctions of the kernel operator defined by the limit of the graph sequence used in the model. This establishes an explicit link between the network structure and the onset of synchronization in the KM on graphs. The results of this work are illustrated with the bifurcation analysis of the KM on Erdos-Renyi, small-world, as well as certain weighted graphs on a circle.
Metrics
Details
- Title
- THE MEAN FIELD ANALYSIS OF THE KURAMOTO MODEL ON GRAPHS II. ASYMPTOTIC STABILITY OF THE INCOHERENT STATE, CENTER MANIFOLD REDUCTION, AND BIFURCATIONS
- Creators
- Hayato Chiba - Kyushu UniversityGeorgi S. Medvedev - Drexel UniversityDepartment of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA
- Publication Details
- Discrete and continuous dynamical systems. Series A, v 39(7), pp 3897-3921
- Publisher
- Amer Inst Mathematical Sciences-Aims
- Number of pages
- 25
- Grant note
- 1715161 / NSF DMS grant; National Science Foundation (NSF); NSF - Directorate for Mathematical & Physical Sciences (MPS)
- Resource Type
- Journal article
- Language
- English
- Academic Unit
- Mathematics
- Web of Science ID
- WOS:000464312800009
- Scopus ID
- 2-s2.0-85065754918
- Other Identifier
- 991019170139304721
InCites Highlights
Data related to this publication, from InCites Benchmarking & Analytics tool:
- Collaboration types
- Domestic collaboration
- International collaboration
- Web of Science research areas
- Mathematics
- Mathematics, Applied