Séminaire Orléans
The Kuramoto Model on GraphsGeorgi Medvedev (Drexel University)
Thursday 23 January 2025 14:00 - Orléans - Salle de Séminaires
Résumé :
In the 1970s, the Japanese statistical physicist Yoshiki Kuramoto demonstrated that a system of all-to-all coupled phase oscillators with random intrinsic frequencies exhibits a phase transition from chaotic (mixing) behavior to synchronization. Moreover, he derived a simple formula for the critical value of the control parameter in this model. The Kuramoto model (KM) has since been widely used to study collective dynamics in various contexts, including neuroscience, population dynamics, and power grids.
The mathematical description of the transition to synchronization in the KM focuses on the bifurcation of the spatially homogeneous solution of the Vlasov PDE. This analysis of this bifurcation turned out to be challenging due to the presence of continuous spectrum on the imaginary axis. The first rigorous study of this bifurcation was completed relatively recently by Hayato Chiba in 2015.
In this talk, I will present joint work with Hayato Chiba (Tohoku University) and Matt Mizuhara (The College of New Jersey) on the KM on graphs. In our work, we explored the effects of different types of connectivity and different distributions of random frequencies of the individual oscillators. These two parameters play a key role in determining the qualitative properties of the spatiotemporal patterns observed in the KM on graphs following the loss of stability of the mixing state. I will discuss the derivation of the mean-field limit of the KM on graphs, as well as the pitchfork and Andronov-Hopf bifurcations in the continuum model and the nascent spatiotemporal patterns.
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