# Synchronization of Chaotic Attractors

Synchronization of Chaotic Attractors

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

2

x

2

x

dy

dt

dz

dt

2

x

where represents prey; and represent two different predators; and , and are positive constants. This generalized Lotka–Volterra equation is augmented by a coupled subsystem

x

y

z

a,b

c

dX

dt

2

X

2

X

d

1

dY

dt

d

2

dZ

dt

2

X

d

3

Here, represents prey; and are predators; and ,, and are coupling parameters. We take with initial conditions and . 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.

X

Y

Z

d

1

d

2

d

3

(a,b,c)=(2.9851,3,2),

y(0)=1.4,

x(0)=y(0)=z(0)=X(0)=Y(0)=Z(0)=1