Generic instability of spatial unfoldings of almost homoclinic periodic orbits

We consider spatially homogeneous time periodic solutions of general partial differential equations. We prove that, when such a solution is close enough to a homoclinic orbit or a homoclinic bifurcation for the differential equation governing the spatially homogeneous solutions of the PDE, then it is generically unstable with respect to large wavelength perturbations. Moreover, the instability is of one of the two following types: either the well-known Kuramoto phase instability, corresponding to a Floquet multiplier becoming larger than I, or a fundamentally different kind of instability, occurring with a period doubling at an intrinsic finite wavelength, and corresponding to a Floquet multiplier becoming smaller than -1.