Abstract

The study of collective synchronization in large ensembles of coupled oscillators has spurred the development of dimensional reduction techniques in nonlinear sciences. We first review the exact method introduced by Watanabe and Strogatz in 1994, which reduces N-dimensional systems of identical phase oscillators to just 3-dimensional ones. Then, we reveal that the conformal automorphisms of the unit disk are actually responsible for such exceptional dynamics. These special Möbius transformations, in close relation with hyperbolic geometry and Poisson kernels, unveil interesting properties of the reduced dynamics. Finally, we simulate the paradigmatic Kuramoto-Sakaguchi model and its extension to two sinusoidally coupled populations, unfolding the so-called chimera states.