Coupled Map Diad Parameter Plane


Coupled map diads are coupled map lattices with only two lattice sites. Nonetheless, they exhibit strikingly complex behavior. These pictures are parameter plane representations of the behavior of two coupled logistic maps, each of which would be chaotic if they were uncoupled. The equations of the system are

Xn+1 = a R Xn(1-Xn)+(1-a) S Yn(1-Yn)
Yn+1 = b S Yn(1-Yn)+(1-b) R Xn(1-Xn)


Chaos and Periodicity

This is the plane of parameters of the coupled logistic map diad with different colors indicating different periodic orbits. Black is period >16 (possibly chaos) blue is period one, green period 2, red period three, etc.













Riddled Parameter Plane

The space of initial conditions for a dynamical system is divided into separate regions, called basins, where one kind of behavior or another is chosen. These basins may be separated by fractal boundaries, or, even worse (or better, depending on whether or not you are delighted by surprising and perverse behavior) there may be riddled basins, when the regions defining different types of behavior are so intimately mixed that regions giving different kinds of bahavior lie arbitrarily close to one another.

If we plot the regions in parameter space for which the system settles into various characteristic behaviors, we may also see riddling.  This is a closeup of the coupled map diad parameter plane in a neighborhood of the riddled area (riddled between period 2 and 3).