- Nonlinear Ordinary Differential Equations

Abstract

These notes are concerned with initial value problems for systems of ordinary differential equations.Here our emphasis will be on nonlinear phenomena and properties, particularly those with physical relevance.Finding a solution to a differential equation may 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 situations 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.Of course, very few nonlinear systems can be solved explicitly, and so one must typically rely on a numerical scheme to accurately approximate the solution.Basic methods for initial value problems, beginning with the simple Euler scheme, and working up to the extremely popular Runge-Kutta fourth order method, will be the subject of the final section of the chapter.However, numerical schemes do not always give accurate results, and we briefly discuss the class of stiff differential equations, which present a more serious challenge to numerical analysts.Without some basic theoretical understanding of the nature of solutions, equilibrium points, and stability properties, 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 [2,9,13,15,17] can be profitably consulted. First Order Systems of Ordinary Differential Equations.Let us begin by introducing the basic object of study in discrete dynamics: the initial value problem for a first order system of ordinary differential equations.Many physical applications lead to higher order systems of ordinary differential equations, but there is a simple reformulation that will convert them into equivalent first order systems.Thus, we do not lose any generality by restricting our attention to the first order case throughout.Moreover, numerical solution schemes for higher order initial value problems are entirely based on their reformulation as first order systems.

The authors' abstract, as published at the source., 2013 · DOI ↗