Review of chaos communication by feedback control of symbolic dynamics

This paper is meant to serve as a tutorial describing the link between symbolic dynamics as a description of a chaotic attractor, and how to use control of chaos to manipulate the corresponding symbolic dynamics to transmit an information bearing signal. We use the Lorenz attractor, in the form of the discrete successive maxima map of the z-variable time-series, as our main example. For the first time, here, we use this oscillator as a chaotic signal carrier. We review the many previously developed issues necessary to create a working control of symbol dynamics system. These include a brief review of the theory of symbol dynamics, and how they arise from the flow of a differential equation. We also discuss the role of the (symbol dynamics) generating partition, the difficulty of finding such partitions, which is an open problem for most dynamical systems, and a newly developed algorithm to find the generating partition which relies just on knowing a large set of periodic orbits. We also discuss the importance of using a generating partition in terms of considering the possibility of using some other arbitrary partition, with discussion of consequences both generally to characterizing the system, and also specifically to communicating on chaotic signal carriers. Also, of practical importance, we review the necessary feedback-control issues to force the flow of a chaotic differential equation to carry a desired message.