Generic instability of spatial unfoldings of almost homoclinic periodic orbits (Q5940480)

This paper deals with the PDEs of the form (1) \(\partial_tu= F(x,\partial_x)\) that is invariant with respect to translations of time (autonomous) and space. The author assumes that \(u\) is in \(\mathbb{R}^d\), \(d\geq 1\), and that the space coordinate \(x\) belongs to \(\mathbb{R}^n\), \(n\geq 1\), or to a domain of \(\mathbb{R}^n\) with boundary conditions of type Neumann or periodic. The author proves that, when spatially homogeneous time periodic solutions of (1) 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.