Deterministic Chaos

Synchronization of Chaotic Processes

Some physical measurements are the average of the measurements of microscopic quantities.

Think of how the temperature of a bath is related to the energy of the individual water molecules, for example.

Suppose instead of the individual logistic maps xⁱ_t, we see only their average value z_t at each time step t,

z_t = (x¹_t + ... + x^N_t)/N

We shall drive an IFS with the sequence of these averages

z₁, z₂, z₃, ...

Though there are many possibilities, we consider only a simple example:

two logistic maps, both with s = 4.

Here the coupling formula becomes

x1t+1	=	(1-c)L(x1t) + cL(x2t)
x2t+1	=	(1-c)L(x2t) + cL(x1t)

We shall use driven IFS and return maps to discover some coupling values where these two chaotic logistic maps synchronize.

Return to Synchronization of Chaotic Processes.