Communication systems via chaotic signals from a reconstruction viewpoint

In this paper, we study communication systems, modulators, and demodulators using chaotic systems from the perspective of reconstruction of attractors. We consider a two-step approach to the recovery of information signals from chaotic modulation. To recover the information signal modulated by a chaotic dynamical system, the first step is to reconstruct the chaotic attractor given a subset of the state variables. The second step is to reconstruct the information signal from the chaotic attractors. Two main techniques to accomplish this are the Takens reconstruction theorem and the chaotic synchronization theorem. In general, a combination of these two techniques are needed to recover the information signal. We show a systematic way to reconstruct an attractor and the injected signal exactly from a scalar observable for a subclass of Lur'e systems. We illustrate this approach using several examples. Extensions to discrete-time systems are also presented.