Genericity and Crystallographic Pinning in Lattice Differential Equations

John Mallet-Paret, Division of Applied Mathematics, Brown University

Lattice differential equations are dynamical systems which are discrete in space and continuous in time. Besides their interest as models for various applications, they are the source of many important mathematical phenomena, such as pattern formation, spatial chaos, and traveling fronts. Of particular interest is the phenomenon of ``pinning,'' in which a traveling front ceases to move due to the spatial inhomogeneity of the lattice. In higher dimensional problems the anisotropy of the lattice can give rise to direction-dependent pinning, so-called ``crystallographic pinning.'' We shall discuss the mathematical foundations of pinning, showing how crystallographic pinning is characterized by an infinite system of transition layer equations. We also show that for the lattice $Z^2$, crystallographic pinning is generic, holding for an open dense set of nonlinearities, and is characterized by certain homoclinic cycles of the layer equations. For higher-dimensional lattices such as $Z^n$, one must generically consider (n-1)-heteroclinic cycles. =====