  The Kuramoto model is a paradigmatic tool for studying the dynamics of
collective behavior in large ensembles of coupled dynamical systems. Over the
past decade a great deal of progress has been made in analytical descriptions
of the macroscopic dynamics of the Kuramoto mode, facilitated by the discovery
of Ott and Antonsen's dimensionality reduction method. However, the vast
majority of these works relies on a critical assumption where the oscillators'
natural frequencies are drawn from a Cauchy, or Lorentzian, distribution, which
allows for a convenient closure of the evolution equations from the
dimensionality reduction. In this paper we investigate the low dimensional
dynamics that emerge from a broader family of natural frequency distributions,
in particular a family of rational distribution functions. We show that, as the
polynomials that characterize the frequency distribution increase in order, the
low dimensional evolution equations become more complicated, but nonetheless
the system dynamics remain simple, displaying a transition from incoherence to
partial synchronization at a critical coupling strength. Using the low
dimensional equations we analytically calculate the critical coupling strength
corresponding to the onset of synchronization and investigate the scaling
properties of the order parameter near the onset of synchronization. These
results agree with calculations from Kuramoto's original self-consistency
framework, but we emphasize that the low dimensional equations approach used
here allows for a true stability analysis categorizing the bifurcations.
