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The goal of these notes is to give a reasonahly com plete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to spe cific problems, including stability calculations. Historical ly, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942. Although the term "Poincare Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it. Hopf's crucial contribution was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds. The method of Ruelle Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations: see for example, Hale [1,2] and Hartman [1]. Of course, other methods are also available. In an attempt to keep the picture balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kopell) of Hopf's original (and generally unavailable) paper.
An understanding of the intricacies in the turbulent combustion process may be a key to solving many of the current energy and environmental problems. The essential nature of turbulent combustion can be derived from the interaction between stochastic flow fluctuations and deterministic molecular processes, such as chemical reaction and transport processes. Undoubtedly, this is one of the most challenging fields of engineering science today, requiring as it does the interaction of scientists and engineers in the respective fields of chemical kinetics and fluid mechanics. The 28 papers in this volume review recent advances in these two disciplines providing new insights into the fundamental processes, addressing a great deal of recent progress. This progress ranges from descriptions of elementary chemical kinetics, to working those descriptions into combustion calculations with large numbers of elementary steps, to improved understanding of turbulent reacting flows and advances in simulations of turbulent combustion. The contributions will inspire further research on many fronts, advancing the understanding of combustion processes, as well as fostering a growing interdisciplinary cooperation.