Computer Algebra Recipes

1.2. THREE-DIMENSIONAL AUTONOMOUS SYSTEMS 41
of X, Y , Z are
_X = k1 AY ¡ k2 XY + k3 AX ¡ k4 X 2 ;
_Y = ¡k1 AY ¡ k2 XY + hk5 Z;
_Z =2k3 AX ¡ k5 Z:
(1.14)
In writing down equations (1.14), use has been made of the following empirical
rule: When two substances react to produce a third, the reaction rate is proportional
to the product of the concentrations of the two substances. Thus, for
example, the structure of the _ X equation can be easily understood. In the ¯rst
chemical reaction, the rate of producing X is +k1 AY. In the second reaction,
there is a decrease in X, the rate contribution being ¡k2 XY. The third reaction
provides a positive contribution +k3 AX. Finally, noting that 2 X in the
fourth reaction is treated as X + X, this reaction provides a rate decrease given
by ¡k4 X 2 . The other rate equations may be similarly understood. The factor
of 2 in the _ Z equation appears because in the third reaction, two of Z appear
for each net (2 X ¡ X) oneofX.
The nonlinear rate equations (1.14) can be converted into a normalized
form that reduces the number of parameters. Introducing a normalized time
¿ =(k1A) t and concentrations
x =(k2X)=(k1 A); y =(k2Y )=(k3 A); z =(k2k5 Z)=(2 k1 k3 A 2 );
and positive parameters,
² = k1=k3; p =(k1A)=k5; q =(k1k4)=(k2 k3);
the Oregonator equations reduce to
² _x(¿) =x + y ¡ qx 2 ¡ xy; _y(¿) =¡y +2hz¡ xy; p _z(¿) =x ¡ z:
As an illustrative example, we will look at the onset of oscillations in the
Oregonator model for ² =0:03, p =2,q =0:006, h =0:75 and initial (normalized)
concentrations x(0) = 1, y(0) = 1, and z(0) = 20.
After loading the plots package,
> restart: with(plots):
the three governing di®erential equations are entered,
> de1:=epsilon*diff(x(t),t)=x(t)+y(t)-q*x(t)^2-x(t)*y(t);
μ
d
de1 := ² x (t) = x(t)+y(t) ¡ q x (t)
dt 2 ¡ x(t) y(t)
> de2:=diff(y(t),t)=-y(t)+2*h*z(t)-x(t)*y(t);
de2 := d
y(t) =¡y(t)+2h z(t) ¡ x (t) y(t)
dt
> de3:=p*diff(z(t),t)=x(t)-z(t);
μ
d
de3 := p
dt z(t)
= x(t) ¡ z(t)