Abstract

The study of the chaotic dynamics in fractional‐order discrete‐time systems has received great attention over the last years. Some efforts have been also devoted to analyze fractional maps with special features. This paper makes a contribution to the topic by introducing a new fractional map that is characterized by both particular dynamic behaviors and specific properties related to the system equilibria. In particular, the conceived one dimensional map is algebraically simpler than all the proposed fractional maps in the literature. Using numerical simulation, we investigate the dynamic and complexity of the fractional map. The results indicate that the new one‐dimensional fractional map displays various types of coexisting attractors. The approximate entropy is used to observe the changes in the sequence sequence complexity when the fractional order and system parameter. Finally, the fractional map is applied to the problem of encrypting electrophysiological signals. For the encryption process, random numbers were generated using the values of the fractional map. Some statistical tests are given to show the performance of the encryption.

Highlights

  • Fractional calculus is a topic which is developed more than 300 years

  • Coexistence of attractors is a special phenomenon in nonlinear dynamical systems, which denotes that with fixed values of system parameters, a tiny disturbance in the initial condition can lead to the coexistence of different attractors. is property makes the chaotic maps very useful in the fields of secure communication and encryption. Since such phenomenon has not received enough attention with fractional discrete-time systems [21], this paper aims to make a contribution by introducing a new fractional map that is characterized by both particular dynamic behaviors and specific properties related to the system equilibria

  • Referring to fractional-order discrete-time systems with special features, this paper has introduced the first example of a fractional map with infinite number of equilibria in a bounded domain. e conceived map has shown coexistence of different types of periodic and chaotic attractors

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Summary

Introduction

Fractional calculus is a topic which is developed more than 300 years. it is only the last decades that it has been extensively and intensively investigated, due to its wide application in signal mechanical controls and other fields [1]. In [10] the hyperchaotic dynamic of the fractional generalized Henon map has been investigated, whereas in [11] the presence of chaos in the fractional discrete memristor system has been illustrated. Erefore, several efforts have been devoted to the study of fractional chaotic maps with some special features related to the system equilibria [14] Among these studies, Zambrano-Serrano et al [15] analyzed the dynamic properties and projective synchronization of the fractional difference map with no equilibrium, whereas in [16] Almatroud et al found rich chaotic behaviours of a novel two-dimensional (2D) hyperchaotic fractional map with infinite line of equilibrium As a result, the analysis of chaotic dynamical behaviours of the fractional-order discrete-time systems without equilibrium points is an interesting topic

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