  Chimera states in spatially extended networks of oscillators have some
oscillators synchronised while the remainder are asynchronous. These states
have primarily been studied in networks with nonlocal coupling, and more
recently in networks with global coupling. Here we present three networks with
only local coupling (diffusive, to nearest neighbours) which are numerically
found to support chimera states. One of the networks is analysed using a
self-consistency argument in the continuum limit, and this is used to find the
boundaries of existence of a chimera state in parameter space.
