WOLFRAM|DEMONSTRATIONS PROJECT

A Chaotic Chemical Reaction Scheme Derived from Chua's Circuit Equations

​
time t
130.
parameters
α
10.
β
1.
γ
16.
initial values
x
0
9.893
y
0
9.945
z
0
9.991
This Demonstration plots the solution of Chua's equations describing a chemical reaction scheme analogous to Chua's electrical circuit. Chua's circuit is the simplest electronic circuit that exhibits classic chaotic behavior [1].
dX
dt
=α(Y-G(X))
,
dX
dt
=β(X-Y+Z)
,
dZ
dt
=-γY
,
where
t
is time;
α
,
β
,
γ
,
a
, and
c
are parameters; and
G(X)=a
3
X
+cX
is a cubic equation with
ac<0
,
a≠0
. The solution to these equations allows both positive and negative values of the variables. A linear transformation of variables can convert this electrical system into a dynamically related chemical scheme that allows only positive state variables [2].
Thus with
x=X+δ
,
y=Y+δ
,
z=Z+δ
, the system becomes:
dx
dt
=α(y-δ)-
3
(x-δ)
-c(x-δ)
,
dy
dt
=β(x-y+z-δ)
,
dz
dt
=-γ(y-δ)
,
where
x
,
y
,
z
represent concentrations of chemical species, with
c=-0.143
,
δ=10
. The chemical reaction scheme that leads to these equations is detailed in Table 1 of [2].