From our work we obtain mixed results. On the one hand, we see that fairly weakly coupled quadratic maps do synchronize, over a large range of initial conditions. On the other, we realize that the analysis of this system requires more potent techniques. One possibility is the statistical analysis of the dynamics. If we knew the average value of , then we could reasonably use it to replace the extremal values of in the above inequality.
The appearance of synchronization is encouraging when we consider the application of coupling to more complicated maps. A good candidate for this sort of analysis is the Ikeda map, which arises when examining properties of nonlinear optical ring cavities. The Ikeda map is complex valued, and given by:
Coupling of this system would physically correspond to two separate optical cavities being linked via a beam splitter or similar device. This method seems to have the potential to reduce chaos in a wide class of dynamical systems, but we are still working on understanding the phenomenon.