Introduction to Stability and Bifurcation in Dynamical Systems and Fluid Mechanics

Abstract

Suppose we are studying a physical system whose state x is governed by an evolution equation $\frac{{dx}}{{dt}} = X\left( x \right)$ which has unique integral curves. Let x₀ be a fixed point of the flow of X; i.e., X(x₀) = 0. Imagine that we perform an experiment upon the system at time t = 0 and conclude that it is then in state x₀. Are we justified in predicting that the system will remain at x₀ for all future time? The mathematical answer to this question is obviously yes, but unfortunately it is probably not the question we really wished to ask. Experiments in real life seldom yield exact answers to our idealized models, so in most cases we will have to ask whether the system will remain near x₀ if it started near x₀. The answer to the revised question is not always yes, but even so, by examining the evolution equation at hand more minutely, one can sometimes make predictions about the future behavior of a system starting near x₀. A trivial example will illustrate some of the problems involved.