Since Kuramoto proposed a model of coupled oscillators, the study of synchronization has pulled the attention from different point of views: biology, chemistry, neuroscience, etc. Such a phenomenon is observed very often in biological systems like the flashing of fireflies, the beating of heart cells and the synaptic firing of neurons in the brain. In this talk we review the state of the art for the Kuramoto model and its associated kinetic counterpart, namely, the Kuramoto—Sakaguchi equation.
The heart of our talk will be to introduce new quantitative estimates on the rate of convergence to the global equilibrium for solutions of the Kuramoto—Sakaguchi equation departing from generic initial data in a large coupling strength regime. Although some previous attempts have been presented for oscillators confined to a half circle, this is the first result in the literature that covers generic initial data, to the best of our knowledge. Notice that the lack of convexity of the problem and the presence of heterogeneities prevent us from using classical methods for gradient flows. Then, the key step will be to introduce a new transportation distance that endows the system with the appropriate structure. In particular, we show new generalized local logarithmic Sobolev and Talagrand type inequalities, similar to the ones derived by Otto and Villani, that quantity convergence to equilibrium.
