WOLFRAM|DEMONSTRATIONS PROJECT

Chemical Reactions Described by the Lorenz Equations

​
time
10.
x
0
32.
y
0
32.
z
0
32.
This Demonstration analyzes the behavior of a chemical reaction scheme that is described by the Lorenz equations:
dX
dt
=a(Y-X)
,
dY
dt
=bX-Y-XZ
,
dZ
dt
=XY-cZ
.
Parameters that lead to interesting behavior are
(a,b,c)=(10,28,8/3)
; the strange attractor that evolves from these equations spans both positive and negative values of
X
and
Y
. If we interpret these symbols as representing concentrations of chemical species, they cannot be negative; a shift of the
X
and
Y
axes can give new variables that are always positive [1]. Thus if we choose
x=X+δ
,
y=Y+δ
,
z=Z
with
δ=30
, the equations describing the system become
dx
dt
=-αx+αy
,
dy
dt
=bx-y+δz-xz-δ(b-1)
,
dz
dt
=-δx-δy-cz+xy+
2
δ
,where
x
,
y
, and
z
represent concentrations of chemical species. Using the values for the constants given above, the equations are
dx
dt
=-10x+10y
,
dy
dt
=28x-y+30z-xz-810
,
dz
dt
=-30x-30y-
8
3
z+xy+900
.
The reactions required to give the Lorenz equations are shown in section (1.7) of [1]. You can vary the time and the initial values of the species to see the evolution of the system. There is an unstable steady state at
(x,y,z)=(38.485,38.485,27)
and an unstable fixed point at
(x,y,z)=
72
+30,
72
+30,27
.