Computer Algebra Recipes

1.1. PHASE-PLANE PORTRAITS 15
The phase-plane analysis of ODE systems, particularly those that are nonlinear,
can be aided by ¯rst locating the stationary or ¯xed points (X0,Y0) of
the system and then identifying their topological nature, i.e., the characteristic
shape of the trajectories in the vicinity of the ¯xed points. At these points all
the derivatives are zero. Thus, from equation (1.1) they are found by solving
P (X0;Y0) =0; Q(X0;Y0) =0: (1.7)
At a stationary point, it follows from equation (1.6) that the slope of the trajectory
is of the form 0=0 and thus indeterminate. At any other point (referred
to as an ordinary point) of the phase plane, a trajectory has a de¯nite slope.
For linear systems that can be expressed in standard form there will be only
one stationary point. For example, setting P = Q =0fortheSHOyieldsa
stationary point at the origin of the v versus x phase plane.
For nonlinear systems, there can be more than one ¯xed point. As an
example, let's use Maple to determine these points for the rabbits{foxes system.
> restart:
The forms of P and Q for equations (1.2) are entered,
> P:=2*r-0.04*r*f;
> Q:=-f+0.01*r*f;
P := 2 r ¡ 0:04 rf
Q := ¡f +0:01 rf
and the solve command applied to the set of equations 1 P =0,Q =0. A
Maple \set" is enclosed in \curly" (f g) brackets. Unlike a Maple list, the
order of the items is not preserved in a Maple set. The unknowns r and f are
also entered as a set.
> solve(fP=0,Q=0g,fr,fg);
fr =0:; f =0:g; fr =100:; f =50:g
From the output, we see that there are two stationary points, one at the origin
of the f-r phase plane and the other at r =100rabbits,f =50foxes.
Every stationary point has a certain topology in its neighborhood that dictates
the nature of the phase-plane trajectories near that point. Thus, identifying
the nature of each stationary point was an important task historically, since
it allowed investigators in the precomputer era to sketch all the possible phase{
plane trajectories from a knowledge of the location of the stationary points
and their types. In the modern computer era we can let software packages like
Maple do the graphing and analysis for us.
For ODE systems that can be put into standard form, what types of stationary
points are possible? It turns out that there are only four types of so-called
simple 2 stationary points, which are schematically illustrated in Figure 1.1. For
the ¯rst three types (the vortex, focal, and nodal points) we shall
1 Note that it isn't necessary to explicitly set P and Q equal to zero, since Maple will
automatically assume that this is so unless you specify otherwise.
2 The precise de¯nition of the phrase \simple" will be given in Chapter 2.