Patterns of an approximately periodic nature appear everywhere. One sees them in cloud formations, on the surface of the saguaro cactus, in the ripples of sand in deserts and near the edge of shallow water, in ocean gravity waves. These lectures are primarily directed toward patterns which arise when a system is driven away from some equilibrium state and undergoes sudden transitions which, with increasing external stress, break the symmetry of the simple, least stressed state, in more and more complicated ways. From such a series of bifurcations, one obtains organized structures which in some sense represent optimal configurations for the system at given values of its intrinsic parameters and the external stress parameter. 202For most of these lectures, we will use convection in fluids, either driven by temperature differences of by electric forces, as our working model. This model, together with the analysis and language which emerges from its investigation, serves as a paradigm for analyzing similar behavior in a broad range of situations. I will also use simple phenomenological models which, although they are not direct reductions of any set of equations which describe accurately the microscopic behavior of a real system, are nevertheless useful because one can work out all the mathematical details simply and explicitly and thereby avoid clouding the main ideas with cumbersome calculations.
|Original language||English (US)|
|Title of host publication||Pattern Formation in the Physical and Biological Sciences|
|Number of pages||68|
|State||Published - Jan 1 2018|
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