Differential Equations: An Introduction to Modern Methods and Applications, 2nd Edition

Nonlinear Differential Equations and Stability

n this chapter, we take up the investigation of systems of nonlinear equations. Such systems can be solved by analytical methods only in rare instances. Numerical approximation methods provide one means of dealing with nonlinear systems. Another approach, presented in this chapter, is geometrical in character and leads to a qualitative understanding of the behavior of solutions rather than to detailed quantitative information. A combination of methods is often needed to achieve optimal results.

7.1 Autonomous Systems and Stability

We first introduced two-dimensional systems of the form

in Section 3.6. Recall that the system (1) is called autonomous because the functions F and G do not depend on the independent variable t. In Chapter 3, we were mainly concerned with showing how to find the solutions of homogeneous linear systems, and we presented only a few examples of nonlinear systems. Now we want to focus on the analysis of two-dimensional nonlinear systems of the form (1). Unfortunately, it is only in exceptional cases that solutions can be found by analytical methods. One alternative is to use numerical methods to approximate solutions. ...