We consider solutions which are homogeneous in space, periodic in time, and close to being homoclinic for a partial differential equation. We show that such solutions are generically unstable with respect to large wavelength perturbations, and that the instability can be of two different types: either the well-known Kuramoto phase insta- bility, or a fundamentally different kind of instability, called self-parametric, displaying a period-doubling and an intrinsic wavelength. We also consider the case where the spatial parity symmetry breaks.