08-08-2012, 11:08 AM
Nonlinear Ordinary Differential Equations
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This chapter is concerned with initial value problems for systems of ordinary differ-
ential equations. We have already dealt with the linear case in Chapter 9, and so 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 configura-
tions 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 typ-
ically 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 breifly discuss the class of stiff differential equations, which present a more serious
challenge to numerical analysts.
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.
Stability.
Once a solution to a system of ordinary differential equations has settled down, its
limiting value is an equilibrium solution; this is the content of Proposition 20.15. However,
not all equilibria appear in this fashion. The only steady state solutions that one directly
observes in a physical system are the stable equilibria. Unstable equilibria are hard to
sustain, and will disappear when subjected to even the tiniest perturbation, e.g., a breath
of air, or outside traffic jarring the experimental apparatus. Thus, finding the equilibrium
solutions to a system of ordinary differential equations is only half the battle; one must
then understand their stability properties in order to characterize those that can be realized
in normal physical circumstances.