Periodicity of Timeseries Generated by Logistic Map. Part I

Abstract

In the paper, the periodicity of pseudo-chaotic implementations in fixed-point calculations is studied using the example of a logistic equation. It was established that with 32-bit represented numbers (three bits - the whole part, 29 bits - the fractional part), the maximum length of the observed cycle is L_(max )< 2^14 iterations, and the space of possible states of the chaotic system after the completion of the transition process is limited S≈2^14 different numbers. Histograms of the duration of transient processes preceding the exit of the trajectory into a cyclic orbit are constructed. It was found that the maximum durations of the transition process do not exceed (2Lmax, 4Lmax). The paper also demonstrates and substantiates the expediency of using a dynamic threshold when forming binary sequences based on chaotic numbers using the threshold method. It is shown that the criterion for the balance of the binary representation of chaotic sequences enables the optimal choice of the number of high-order bits that must be discarded in order to obtain a uniform distribution. Approaches to increase the cyclicity of period of digital implementations of chaotic systems are analyzed. It is shown that the period of the external disturbance must be coordinated with the durations of the cycles observed in the chaotic system. The results of the work show the limitations of chaotic systems, which must be correctly taken into account when using them in cryptography.