Chaos Pass Filter: Linear Response of Synchronized Chaotic Systems

TL;DR

This paper investigates how synchronized chaotic systems respond linearly to small external signals, revealing complex behaviors like power law tails, fractal structures, and resonance effects, with implications for chaos pass filtering.

Contribution

It provides analytical and numerical analysis of the linear response of synchronized chaotic systems, including the distribution of deviations and bit error rates, highlighting phenomena like fractal structures and resonance effects.

Findings

Distribution of deviations can have power law tails.

Bit error rate exhibits nonmonotonic behavior and fractal structures.

A chain of units can completely filter out perturbations.

Abstract

The linear response of synchronized time-delayed chaotic systems to small external perturbations, i.e., the phenomenon of chaos pass filter, is investigated for iterated maps. The distribution of distances, i.e., the deviations between two synchronized chaotic units due to external perturbations on the transfered signal, is used as a measure of the linear response. It is calculated numerically and, for some special cases, analytically. Depending on the model parameters this distribution has power law tails in the region of synchronization leading to diverging moments of distances. This is a consequence of multiplicative and additive noise in the corresponding linear equations due to chaos and external perturbations. The linear response can also be quantified by the bit error rate of a transmitted binary message which perturbs the synchronized system. The bit error rate is given by an integral over the distribution of distances and is calculated analytically and numerically. It displays a complex nonmonotonic behavior in the region of synchronization. For special cases the distribution of distances has a fractal structure leading to a devil's staircase for the bit error rate as a function of coupling strength. The response to small harmonic perturbations shows resonances related to coupling and feedback delay times. A bi-directionally coupled chain of three units can completely filtered out the perturbation. Thus the second moment and the bit error rate become zero.