Computer Algebra Recipes

Chapter 1
Phase-Plane Portraits
Every portrait that is painted with feeling
is a portrait of the artist, not of the sitter.
Oscar Wilde, Anglo-Irish playwright, novelist, and poet (1854{1900)
Consider a system of two ¯rst-order coupled ODEs of the general structure
_X ´ dX
= P (X;Y ); _
dY
Y ´ = Q(X;Y ); (1.1)
dt dt
where P and Q are known functions of the dependent variables X and Y ,and
the independent variable has been taken to be the time t. In other model
equations, the independent variable could be a spatial coordinate, e.g., the
Cartesian coordinate x. For compactness, the dot notation of (1.1) will often
be used in our text discussion for time derivatives, one dot denoting d=dt, two
dots standing for d2 =dt2 , and so on. Superscripted primes on the dependent
variable indicate a spatial derivative, e.g., Y 0 ´ dY=dx, Y 00 ´ d2Y=dx2 ,etc.
The 2-dimensional ODE system (1.1) is said to be autonomous, meaningthat
P and Q do not depend explicitly on t. If there is an explicit dependence on the
independent variable, the equations are said to be nonautonomous. Our goal in
the following section is to illustrate a simple graphical procedure for exploring
all possible solutions of equations (1.1) for speci¯c forms of P and Q. In the
second section, we shall show that a 2-dimensional nonautonomous system of
ODEs can be cast into a 3-dimensional autonomous system and present some
interesting examples of the latter.
1.1 Phase-Plane Portraits
Some biological models of competing species are naturally of the standard form
(1.1). For example, a simple ODE model of the temporal evolution of interacting
rabbit and fox populations might be given by the system
_r =2r ¡ 0:04 rf; f _ = ¡f +0:01 rf; (1.2)
where r(t) andf(t) are the numbers of rabbits and foxes per unit area at time
t. If the \interaction" terms involving rf are omitted in (1.2), the remaining
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