# The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations

@article{Chiba2019TheMF, title={The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations}, author={Hayato Chiba and Georgi S. Medvedev}, journal={Discrete \& Continuous Dynamical Systems - A}, year={2019} }

In our previous work [Chiba, Medvedev, arXiv:1612.06493], 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…

## 9 Citations

Bifurcations in the Kuramoto model on graphs.

- MathematicsChaos
- 2018

This work studies several model problems illustrating the link between network topology and synchronization in coupled dynamical systems, and identifies several families of graphs for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in a similar manner.

Stability and Bifurcation of Mixing in the Kuramoto Model with Inertia

- MathematicsSIAM Journal on Mathematical Analysis
- 2022

The Kuramoto model of coupled second order damped oscillators on convergent sequences of graphs is analyzed in this work. The oscillators in this model have random intrinsic frequencies and interact…

The Kuramoto Model on Power Law Graphs: Synchronization and Contrast States

- MathematicsJ. Nonlinear Sci.
- 2020

It is shown that despite sparse connectivity, power law networks possess remarkable synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law distribution.

Pattern Formation in Random Networks Using Graphons

- Mathematics, Computer Science
- 2021

This work uses center manifold theory to characterize Turing bifurcations in the continuum limit in a manner similar to the classical partial differential equation case and derives estimates that relate the eigenvalues and eigenvectors of the finite graph Laplacian to those of the graphon La Placian.

Chimeras Unfolded

- PhysicsJournal of Statistical Physics
- 2022

This paper describes a codimension–2 bifurcation of mixing whose unfolding, in addition to the classical scenario of the onset of synchronization, also explains the formation of clusters and chimeras and shows that network topology can endow chimera states with nontrivial spatial organization.

Mathematical Framework for Breathing Chimera States

- PhysicsJ. Nonlinear Sci.
- 2022

About two decades ago it was discovered that systems of nonlocally coupled oscillators can exhibit unusual symmetry-breaking patterns composed of coherent and incoherent regions. Since then such…

Mean-field limit of non-exchangeable systems

- Mathematics
- 2021

This paper deals with the derivation of the mean-field limit for multi-agent systems on a large class of sparse graphs. More specifically, the case of non-exchangeable multi-agent systems consisting…

Mean field limits of co-evolutionary heterogeneous networks

- Computer Science
- 2022

This work is the first to rigorously address the mean field limit (MFL) of a co-evolutionary network model, which is described by solutions of a generalized Vlasov type equation.

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