Ned Corron

It is known that chaotic oscillators can synchronize when coupled by a scalar channel. This effect has been suggested for a number of applications, including spread-spectrum communications using modulated chaotic waveforms. However, the process of synchronizing nonlinear oscillators is not entirely well understood, and there is some debate as to whether synchronization is a robust process in the presence of a limited channel. Here, we use symbolic dynamics to examine the flow of information in unidirectionally coupled chaotic oscillators exhibiting synchronization. The theory of symbolic dynamics reduces chaos to a shift map that acts on a discrete set of symbols, each of which contains information about the system state. Using this transformation we can establish the minimal channel requirements for high-quality synchronization. We also explore so-called "achronal" synchronization, in which the response lags or leads the drive by a fixed amount of time. We find fundamental tradeoffs between the precision to which the drive state is detected, the quality of synchronization attained, and the delay or anticipation exhibited by the response system. Along the way, physical experiments using electronic circuits are described that illustrate the role of symbolic dynamics and information flow in chaos synchronization.