  Motivated by phenomena related to biological systems such as the
synchronously flashing swarms of fireflies, we investigate a network of phase
oscillators evolving under the generalized Kuramoto model with inertia. A
distance-dependent, spatial coupling between the oscillators is considered.
Zeroth and first order kernel functions with finite kernel radii were chosen to
investigate the effect of local interactions. The hysteretic dynamics of the
synchronization depending on the coupling parameter was analyzed for different
kernel radii. Numerical investigations demonstrate that (1) locally locked
clusters develop for small coupling strength values, (2) the hysteretic
behavior vanishes for small kernel radii, (3) the ratio of the kernel radius
and the maximal distance between the oscillators characterizes the behavior of
the network.
