Over the past decade it has become increasingly clear that various hydrodynamic instabilities can be understood through the analysis of relatively simple nonlinear models that manifest complicated dynamical behavior. These models usually consist of a set of coupled nonlinear differential equations or a set of coupled nonlinear maps. Both these types of models are discussed in the present book. The author adeptly picks his way through the snarled equations of continuous hydrodynamic systems involving fluid flow, temperature fields, density differences, and magnetic fields, pointing out the types of bifurcations here and the similarities in
structure there. He begins with a treatment of the Navier-Stokes equations coupled to an inhomogeneous temperature field, i.e., the Rayleigh-B6nard problem, introduces linear stability theory, and discusses the properties of the linear eigenvalues characterizing the onset of convection. A multiplescale analysis in both space and time is used to describe the fluid behavior near the point of onset, and the instabilities of the convection state are discussed.