| Few people are surprised when periodic phenomena, pendula of
clocks or the steps of marching bands, for example, synchronize. Here we'll see that
under some circumstances, chaotic processes can synchronize. This is surprising,
given the sensitivity to initial conditions exhibited by chaotic systems. |
| To demonstrate this, we'll build networks of coupled
logistic maps. We write logistic maps as
L(s,x) = s⋅x⋅(1 - x) because in some situations the
s-values will be different for each map. Simplest is a network of two maps |
| xn+1 = c11L(s1,xn) + c12L(s2,yn) |
| yn+1 = c21L(s1,xn) + c22L(s2,yn) |
|  |
|
| where the coupling constants satisfy c11 + c12 = 1 and
c21 + c22 = 1. |
| The three map network is |
| xn+1 = c11L(s1,xn) + c12L(s2,yn)
+ c13L(s3,zn) |
| yn+1 = c21L(s1,xn) + c22L(s2,yn)
+ c23L(s3,zn) |
| zn+1 = c31L(s1,xn) + c32L(s2,yn)
+ c33L(s3,zn) |
|  |
|
| with the obvious conditions c11 + c12 + c13 = 1,
c21 + c22 + c23 = 1, and
c31 + c32 + c33 = 1. |
| We can build even larger networks, and set some of the coupling constants to 0, imposing
complicated geometries of who can talk with whom. |
| |
| We could plot the output of each map, but because much of what we see of the world is the
average behavior of many agents, instead we'll plot the average output of these maps, |
| avn = (xn + yn)/2 or
avn = (xn + yn + zn)/3 |
| for example. These are our laboratory for exploring complex behavior and synchronization. |