In Chapter 9 we developed some methods for discovering the behavior of linear autonomous systems of differential equations. While not all methods may have worked on a specific system, we could always determine the nature of their equilibrium points. However, models of many phenomena, such as spring-mass systems, simple pendulums, epidemics, population growth, bungee jumping, and predator-prey systems, often involve nonlinear autonomous systems. These nonlinear systems are more difficult to analyze than linear systems. In this chapter we examine several examples of what may occur and develop a number of approaches that give us qualitative information about the behavior of solutions. We collect these together in a catalog of techniques to help construct a phase portrait for nonlinear systems.

To relate the material in this chapter to that from prior chapters, we discuss some differences between linear and nonlinear systems and between autonomous and nonautonomous systems.