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 am generically unstable with respect to large wavelength perturbations, and that the instability can be of two different types : either the well-known Kuramoto phase instability, 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.
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