Observability-singularity manifolds in the context of chaos based cryptography

In the ´80s Takens formulated the conditions that ensure the capability to reconstruct the dynamics of a transmitter when an observer receives one scalar output from the transmitter. In practical situations, the reconstruction of the original system is strongly influenced by the choice of the variable transmitted over the communication channel. This paper aims to analyze this influence in the context of mathematical singularities occurring in the evolution of the chaotic manifolds used in encryption. We analyze two systems having a chaotic behavior, a discrete system, the Hitzl-Zele map, and a continuous one, the Colpitts oscillator. We show the existence of observability singularities in both cases. The numerical experiments point out that the dynamics of the discrete system falls in these singularities sets, but very infrequently. More surprisingly, the dynamics of the continuous system can not pass through the singularity, which is situated at infinity. But an exponential factor allows the chaotic dynamics to approach the vicinity of the singularity better than 10^-7 and that, for about 30% of its duration. The noise inherent in analog signals are much higher than this value, the observation of the system is impossible in practice.