A smooth chaotic map with parameterized shape and symmetry

Daniel P.B Chaves,Cecilio Pimentel,Carlos E.C Souza

doi:10.1186/s13634-016-0405-4

Daniel P.B Chaves, Cecilio Pimentel + Show 1 more

Open Access

https://doi.org/10.1186/s13634-016-0405-4

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Abstract

We introduce in this paper a new chaotic map with dynamical properties controlled by two free parameters. The map definition is based on the hyperbolic tangent function, so it is called the tanh map. We demonstrate that the Lyapunov exponent of the tanh map is robust, remaining practically unaltered with the variation of its parameters. As the main application, we consider a chaotic communication system based on symbolic dynamics with advantages over current approaches that use piecewise linear maps. In this context, we propose a new measure, namely, the spread rate, to study the local structure of the chaotic dynamics of a one-dimensional chaotic map.

Highlights

Chaotic signals are characterized by irregularity, aperiodicity, decorrelation, and broadband
We focus on chaotic maps with suitable properties for chaotic modulation schemes based on symbolic dynamics [21]
We propose and analyze a one-dimensional chaotic map based on the tangent hyperbolic function with suitable properties for chaotic modulation schemes based on symbolic dynamics [21]

Summary

Introduction

Chaotic signals are characterized by irregularity, aperiodicity, decorrelation, and broadband. It is capable of avoiding problems with error amplification and precision truncation found in the former scheme, and if implemented with adequate maps, it permits to trade performance and security In this context, a chaotic communication system based on a piecewise linear chaotic map with a parameter that controls the length of a guard region (never visited region) was proposed in [24]. The tanh map is specified by two parameters that permit to control the symmetry of the map (even or odd) and its shape The latter has strong implication on the invariant distribution of the tanh map and may be used to generate a seldom visited region that works as a guard region on chaos-based communication schemes.

Chaotic maps

Lyapunov exponent and orbit diagram for the tanh map

Findings

Conclusions