Computer Algebra Recipes

2.1. PHASE-PLANE ANALYSIS 79
> sol2:=[seq(allvalues(sol[i]),i=1..N)];
"
"
p # "
p #
3
3
sol2 := [x =0;y=0]; x =0;y= ; x =0;y= ¡ ;
3
3
" p
5
[x =1;y=0]; [x = ¡1; y=0]; x =
5 ;y=
p #
30
;
15
" p p # " p
5 30
5
x = ;y= ¡ ; x = ¡
5 15
5 ;y=
p # " p p ##
30
5 30
; x = ¡ ;y= ¡
15
5 15
In the sol2 list of lists, there are clearly nine lists of x and y values, corresponding
to the coordinates of nine stationary points. There is indeed a plethora
of ¯xed points in this example. The number of operands (lists here) will be
needed, so this number is extracted with the nops command.
> N2:=nops(sol2);
N2 := 9
Functional operators are formed to numerically evaluate p = ¡(a + d) and
q = ad¡ bc for the ith entry in sol2 .
> p:=i->evalf(eval(-(a+d),sol2[i])):
> q:=i->evalf(eval(a*d-b*c,sol2[i])):
Next, Jennifer creates a do loop, running from i =1toN2 = 9, to identify the
nature of each ¯xed point and to plot each point.
> for i from 1 to N2 do
The ith entry (list of x and y coordinates) of sol2 is entered, and then the x and
y coordinates are extracted separately. For example, taking i =1,sol2[1,1]
extracts the ¯rst entry (x = 0) in the ¯rst list of sol2 , while sol2[1,2] produces
the second entry (y = 0). The right-hand sides are extracted to give the x and
y coordinates of the ¯xed point for plotting purposes.
> sol2[i]; X[i]:=rhs(sol2[i,1]); Y[i]:=rhs(sol2[i,2]);
The values of p, q, andr = p2 ¡ 4 q are determined for the ith ¯xed point. Jennifer
uses the concatenation operator || to attach numbers to these quantities.
For example, the outputs of p||1, p||2, etc., would be of the form p1 , p2 ,etc.
Concatenation is a useful alternative way of numbering quantities.
> p||i:=p(i); q||i:=q(i); r||i:=simplify(p||i^2-4*q||i);
A conditional statement is now entered that will classify the ith ¯xed point
according to the region of the p-q diagram in which it is located. The general
syntax for a conditional statement is
if then
elif then
else
end if
where the elif (else if) and else phrases are optional.