In some applications, the qualitative behavior of the solution, rather than the explicit solution, is of interest. For instance, one could be interested in the determination of whether operating at an equilibrium point is stable or not. In most cases, we may want to see how the different solutions together form a portrait of the behavior around particular neighborhoods of interest. The portraits can show how different points such as sources, sinks, or saddles are interacting to affect neighboring solutions. For most scientific applications, a better understanding of a process requires the larger portrait, including how they would change with variations in critical parameters.
We begin this chapter with a brief summary on the existence and uniqueness of solutions to differential equations. Then we define and discuss the equilibrium points of autonomous sets of differential equations, because these points determine the sinks, sources, or saddles in the solution domain. Next, we explain some of the technical terms, such as integral curves, flows, and trajectories, which are used to define different types of stability around equilibrium points. Specifically, we have Lyapunov stability, quasi-asymptotic stability, and asymptotic stability.
We then briefly investigate the various types of behavior available for a linear second-order system, dx/dt =
Ax, A[=]2 × 2, for example, nodes, focus, and centers. Using the tools provided in previous chapters, we end up with a convenient map that relates the different types of behavior, stable or unstable, to the trace and determinant of A.