Synchronization in the Kuramoto model

I will discuss the synchronization in the Kuramoto model. Starting with the simplest example, which is based on the papers [1,2], I will explain why the Kuramoto model on a complete graph is almost integrable and how the dynamics of this model is related to hyperbolic geometry. Then I will explicitly find the integrals of motion and show how everything mentioned above is related to the Watanabe-Strogatz ansatz. After that, I will show that the Kuramoto model on a star graph also admits dimensional reduction and that it can be solved exactly in the limit of a large number of star rays [3].

References

1. Seth A. Marvel, Renato E. Mirollo, Steven H. Strogatz, Chaos 19, 19 (2009), 043104
2. Bolun Chen, Jan R. Engelbrecht, Renato Mirollo, J. Phys. A: Math. Theor., 50 (2017), 355101, arXiv: 1707.00713
3. Vladimir Vlasov, Yong Zou, Tiago Pereira, Phys. Rev. E, 92 (2015), 012904, arXiv: 1411.6873