Pattern formation in spatially extended systems has been observed recently in a number of experimental systems. These include surface reactions on catalytic ribbons and homogeneous chemical reactions in gels. These patterns do not resemble the spatially periodic structure of Turing patterns but have spatially localized patterns and spatially differentiated dynamics. Moreover, these patterns appear in both PDE systems and in systems of coupled cells. The basis of these new patterns appears to be in non-local non-diffusive coupling. In another direction regular patterns have been observed in the time-averages of spatio-temporally chaotic regimes in the Faraday surface wave experiment and in rotating convection experiments. It is the purpose of this workshop to delve into the mathematical origin of these new types of pattern and the kinds of (numerical and physical) experiments that may exhibit such patterns.

We describe briefly some of the mathematical issues. In recent years, starting with the work of Field and the examples of Guckenheimer and Holmes, it has been seen that the dynamics of systems with symmetry has different structurally stable dynamics than does systems without symmetry. These differences include heteroclinic cycles, drifting along group orbits, symmetric chaos, instant chaos and issues of forced symmetry breaking.

Heteroclinic cycles give one mathematical formulation for intermittency and can be formed in two different ways. The first is through standard bifurcations where saddle-sink connections are formed in invariant (symmetry) planes. The second method involves system symmetry breaking where a perturbation parameter in the equations leads to equations with less symmetry. In this case an orbitally stable group orbit of equilibria is perturbed to a dynamically invariant manifold with less symmetry and more complicated dynamics. In another direction symmetry of attractors has been shown to lead to group invariant time-averages and patterns in time-averages in experiments with spatio-temporal chaos.

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