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.