Synchronization of Chaotic Lorenz Attractors
by David Rodrigues
Lorenz attractors are chaotic, meaning that even if one starts with two very close points, they will diverge into two completely different trajectories.
These three simple equations govern the behaviour of the system, and as such the system is completely deterministic. The problem arises when one really wants to know the result of the integration in the long run... any small error in the early steps of calculation, any perturbation, will result in a completely different trajectory and in a different point in space.
The white line measures the distance between two trajectories as time passes. One can see it oscillates in a non deterministic way never indicating synchronization.
How to synchronize two Lorenz attractors?
Imagine that at each time step you magically connect the X component of the equations above, meaning that the value of X used in the equation for attractor 1 is copied over to attractor 2. Then you compute the X,Y and Z at t+1 for both attractors as usual.
In the above applet what you see this effect. The continuous line is attractor 1 (the reference) and the dashed lines are from the attractor 2 (the coupled attractor), meaning that at each time step the X value from 1 is copied over to 2.
The white line measures the distance between two trajectories as time passes. In this case we can quickly see it dropping down to zero and synchronization occurs.
This changes the behaviour of the system completely. Instead of being two systems with divergent trajectories they get closer and closer until finally they overlap perfectly, travelling in sync through space.