Chaos Pass Filter: Linear Response of Synchronized Chaotic Systems

TL;DR

This paper investigates the linear response of synchronized chaotic systems with delays, analyzing their behavior under small perturbations, and explores implications for secure communication and error rates.

Contribution

It provides analytical and numerical analysis of the response of synchronized chaotic systems to perturbations, including the distribution of distances and bit error rates, with insights into communication security.

Findings

Distribution of distances can have power law tails leading to diverging moments.

Bit error rate can be reduced by adding noise, and is not linked to Lyapunov exponents.

The response exhibits resonances related to delay times and a devil's staircase structure in error rates.

Abstract

The linear response of synchronized chaotic units with delayed couplings and feedback to small external perturbations is investigated in the context of communication with chaos synchronization. For iterated chaotic maps, the distribution of distances is calculated numerically and, for some special cases, analytically as well. Depending on model parameters, this distribution has power law tails leading to diverging moments of distances in the region of synchronization. The corresponding linear equations have multiplicative and additive noise due to perturbations and chaos. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. For perturbation from a binary message the bit error rate is calculated. The bit error rate is not related to the transverse Lyapunov exponents, and it can be reduced when additional noise is added to the transmitted signal. For some special cases, the bit error rate as a function of coupling strength has the structure of a devil's staircase, related to an iterated function system. Finally, the security of communication is discussed by comparing uni- and bi-directional couplings.