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Synchronization of Chaotic Attractors

​
time, t
90
coupling parameters
d
1
0
d
2
0
d
3
0
time plots
This Demonstration illustrates a method for synchronizing the trajectories of two modified Lotka–Volterra systems. Synchronization is accomplished by linear feedback [1], with the governing equations of the first system [2] as follows:
dx
dt
=x-xy+c
2
x
-az
2
x
,​
dy
dt
=-y+xy
,​
dz
dt
=-bz+az
2
x
,
where
x
represents prey;
y
and
z
represent two different predators; and
a,b
, and
c
are positive constants. This generalized Lotka–Volterra equation is augmented by a coupled subsystem
dX
dt
=X-XY+c
2
X
-aZ
2
X
+
d
1
(x-X)
,​
dY
dt
=-Y+XY+
d
2
(y-Y)
,​
dZ
dt
=-bZ+aZ
2
X
+
d
3
(x-X)
.
Here,
X
represents prey;
Y
and
Z
are predators; and
d
1
,
d
2
, and
d
3
are coupling parameters. We take
(a,b,c)=(2.9851,3,2),
with initial conditions
y(0)=1.4,
and
x(0)=y(0)=z(0)=X(0)=Y(0)=Z(0)=1
. When the system is uncoupled (i.e., the coupling parameters are zero), these two systems diverge rapidly; in contrast, when the system is synchronized, the trajectories become superimposed.
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