Title: Mean field repulsive Kuramoto models: Phase locking and spatial signs,
Abstract: The phenomenon of self-synchronization in populations of oscillatory units
appears naturally in neurosciences. However, in some situations, the formation
of a coherent state is damaging. In this article we study a repulsive
mean-field Kuramoto model that describes the time evolution of n points on the
unit circle, which are transformed into incoherent phase-locked states. It has
been recently shown that such systems can be reduced to a three-dimensional
system of ordinary differential equations, whose mathematical structure is
strongly related to hyperbolic geometry. The orbits of the Kuramoto dynamical
system are then described by a ow of Möbius transformations. We show this
underlying dynamic performs statistical inference by computing dynamically
M-estimates of scatter matrices. We also describe the limiting phase-locked
states for random initial conditions using Tyler's transformation matrix.
Moreover, we show the repulsive Kuramoto model performs dynamically not only
robust covariance matrix estimation, but also data processing: the initial
configuration of the n points is transformed by the dynamic into a limiting
phase-locked state that surprisingly equals the spatial signs from
nonparametric statistics. That makes the sign empirical covariance matrix to
equal 1 2 id2, the variance-covariance matrix of a random vector that is
uniformly distributed on the unit circle.