Dynamics of coding in communicating with chaos

Recent work has considered the possibility of utilizing symbolic representations of controlled chaotic orbits for communicating with chaotically behaving signal generators. The success of this type of nonlinear digital communication scheme relies on partitioning the phase space properly so that a good symbolic dynamics can be defined. A central problem is then how to encode an arbitrary message into the wave form generated by the chaotic oscillator, based on the symbolic dynamics. We argue that, in general, a coding scheme for communication leads to, in the phase space, restricted chaotic trajectories that live on nonattracting chaotic saddles embedded in the chaotic attractor. The symbolic dynamics of the chaotic saddle can be robust against noise when the saddle has large noise-resisting gaps covering the phase-space partition. Nevertheless, the topological entropy of such a chaotic-saddle, or the channel capacity in utilizing the saddle for communication, is often less than that of the chaotic attractor. We present numerical evidences and theoretical analyses that indicate that the channel capacity associated with the chaotic saddle is generally a nonincreasing, devil's-staircase-like function of the noise-resisting strength. There is usually a range for the noise strength in which the channel capacity decreases only slightly from that of the chaotic attractor. The main conclusion is that nonlinear digital communication using chaos can yield a substantial channel capacity even in noisy environment.