We discuss recent results in the theory of lattice dynamical systems, namely infinite-dimensional dynamical systems which possess a discrete spatial structure modeled on a (usually high-dimensional) lattice. Typical systems incorporate both local nonlinear dynamics and interactions between nearby points of the lattice in a nontrivial way. Of particular interest are stable equilibria with the resulting displays of either regular patterns or spatial chaos, traveling wave solutions, propagation failure, the effects of anisotropy, and associated bifurcation phenomena.

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1996-1997 Mathematics in High Performance Computing