Is the Ott-Antonsen manifold attracting?

Abstract

The Kuramoto model is a paradigm for studying oscillator networks with interplay between coupling tending towards synchronization, and heterogeneity in the oscillator population driving away from synchrony. In continuum versions of this model an oscillator population is represented by a probability density on the circle. Ott and Antonsen identified a special class of densities which is invariant under the dynamics and on which the dynamics are low-dimensional and analytically tractable. The reduction to the OA manifold has been used to analyze the dynamics of many variants of the Kuramoto model. To address the fundamental question of whether the OA manifold is attracting, we develop a systematic technique using weighted averages of Poisson measures for analyzing dynamics off the OA manifold. We show that for models with a finite number of populations, the OA manifold is {\it not} attracting in any sense; moreover, the dynamics off the OA manifold is often more complex than on the OA manifold, even at the level of macroscopic order parameters. The OA manifold consists of Poisson densities $\rho_\omega$. A simple extension of the OA manifold consists of averages of pairs of Poisson densities; then the hyperbolic distance between the centroids of each Poisson pair is a dynamical invariant (for each $\omega$). These conserved quantities, defined on the double Poisson manifold, are a measure of the distance to the OA manifold. This invariance implies that chimera states, which have some but not all populations in sync, can never be stable in the full state space, even if stable in the OA manifold. More broadly, our framework facilitates the analysis of multi-population continuum Kuramoto networks beyond the restrictions of the OA manifold, and has the potential to reveal more intricate dynamical behavior than has previously been observed for these networks.