A plain man's guide to bifurcations

Bifurcation theory, in its broad sense, includes a large body of theory and techniques for understanding nonlinear systems. Its uses range from deep theoretical understanding (and even wild speculation) to practical engineering design. Its broad applicability is because it deals with qualitative changes such as symmetry breaking as in the buckling of a beam, or appearance of new behavioral modes as in the development of instability in a railway axle. Naturally, many of its results have been discovered independently by workers in different fields, and one reason for writing this introduction is to persuade researchers who may have already used some of its results to learn something of the general theory, and perhaps to contribute their own knowledge to the field.