Computer Algebra Recipes

1.2. THREE-DIMENSIONAL AUTONOMOUS SYSTEMS 43
(initial condition), the trajectory will wind onto the same loop, indicating that
the loop is a stable 3-dimensional limit cycle.
The odeplot command is used to plot the concentrations x(t), y(t), and z(t)
as red, blue, and green curves respectively, with a title indicating this included.
> odeplot(sol,[[t,x(t),color=red],[t,y(t),color=blue],
[t,z(t),color=green]],0..20,numpoints=3000,tickmarks=[3,3],
thickness=2,title="red=x, blue=y, green=z");
x, y, z
100
50
0
red=x, blue=y, green=z
10 20
t
Figure 1.17: Oscillatory behavior of HBrO2 (x), Br ¡ (y), and Ce 4+ (z).
The black-and-white version is shown in Figure 1.17, the tallest curve being x(t),
the intermediate curve z(t), and the shortest curve y(t). Each curve reaches its
maximum amplitude at a di®erent time.
PROBLEMS: Problem 1-18: Oregonator limit cycle
Con¯rm that a limit cycle results in the Oregonator model, regardless of the
initial (nonzero) concentrations.
Problem 1-19: Another chemical oscillator
The rate equations for a certain chemical oscillator are
A k1
! X
B + X k2
! Y + ¤
2 X + Y
k3
! 3 X
X k4
! ¤
where the concentrations A and B of species A and B are held constant.
(a) Using the empirical rule for chemical reactions, write down the rate equations
for X and Y .