Nonlinear Ordinary Differential Equations

Most physical processes are modeled by differential equations. First order ordinary differential equations, also known as dynamical systems, arise in a wide range of applications, including population dynamics, mechanical systems, planetary motion, ecology, chemical diffusion, etc., etc. See [19, 72,ODES] for additional material and applications. The goal of this chapter is to study and solve initial value problems for nonlinear systems of ordinary differential equations. Of course, very few nonlinear systems can be solved explicitly, and so one must typically rely on a suitable numerical scheme in order to approximate the solution. However, numerical schemes do not always give accurate results. Without some basic theoretical understanding of the nature of solutions, equilibrium points, and their stability, one would not be able to understand when numerical solutions (even those provided by standard well-used packages) are to be trusted. Moreover, when testing a numerical scheme, it helps to have already assembled a repertoire of nonlinear problems in which one already knows one or more explicit analytic solutions. Further tests and theoretical results can be based on first integrals (also known as conservation laws) or, more generally, Lyapunov functions. Although we have only space to touch on these topics briefly, but, we hope, this will whet the reader’s appetite for delving into this subject in more depth. The references [19, 46, 72, 80, 85] can be profitably consulted. Our overriding emphasis will be on those properties of solutions that have physical relevance. Finding a solution to a differential equation is not be so important if that solution never appears in the physical model represented by the system, or is only realized in exceptional circumstances. Thus, equilibrium solutions, which correspond to configurations in which the physical system does not move, only occur in everyday a physics if they are stable. An unstable equilibrium will not appear in practice, since slight perturbations in the system or its physical surroundings will immediately dislodge the system far away from equilibrium. Finally, we present a few of the most basic numerical solution techniques for ordinary differential equations. We begin with the most basic Euler scheme, and work up to the Runge–Kutta fourth order method, which is one of the most popular methods for everyday applications.