Mathematics of Synchronisation

When there is a big applause after a good concert, sometimes the audience starts to clap in a synchronised manner even though there is no one synchronising the different people. Similarly, in biology, the actions of individual neural cells in the heart are synchronised to create a joint periodic impulse.

These examples can be modelled as a system of coupled oscillators. This leads to the Kuramoto model, where the evolution of each oscillator is determined by its natural frequency and the coupling. On the one hand, the coupling works towards a synchronisation. On the other hand, the natural frequencies differ, which desynchronises the system. Using the mean-field limit, a large number of oscillators can be described by a density over the phase space. Its evolution is then described by a PDE, which is a non-linear transport equation without any dissipation.

Despite the lack of dissipation, stable stationary states are observed in numerical simulations. Here, the stability is due to the frequency differences and can be understood as phase mixing. This is the same stability mechanism as in Landau damping for the Vlasov-Poisson equation or inviscid damping for the Euler equation. It is a delicate stability mechanism because there is no stability in strong topology but only in weak topology.

In this talk, I will introduce the topic and describe this fascinating stability mechanism. Then I will present my results on the stability of inhomogeneous stationary states and explain the main ideas of the proof.