Topics in Stability and Bifurcation Theory

In analyzing the dynamics of a physical system governed by nonlinear equations the following questions present themselves: Are there equilibrium states of the system? How many are there? Are they stable or unstable? What happens as external parametera are varied? As the parameters are varied, a given equilibrium may lose its stability (although it may continue to exist as a mathematical solution of the problem) and other equilibria or time periodic oscillations may branch off. Thus, bifurcation is a phenomenon closely related to the loss of stability in nonlinoar physical systems.The subjects of bifurcation and stability have always attracted the interest of pure mathematicians, beginning at least with Poincare and Lyapounov. In the past decade an increasing amount of attention пае focused on problems in partial differential equations. The purpose of these notes is to present some of the basic mathematical mothods which have developed during this period. They are primarily mathematical in their approach, but it is hoped they will be of value to thoae applied mathematicians and engineers interested in learning the mathematical techniques of the subject.