Symmetries in Hidden Bifurcation Routes to Multiscroll Chaotic Attractors Generated by Saturated Function Series

Faiza Zaamoune,René Lozi,Guanrong Chen,Tidjani Menacer

doi:10.25073/jaec.201934.256

Abstract

In this paper, hidden bifurcation routes to multiscroll chaotic attractors generated by saturated function series are explored. The method to nd such hidden bifurcation routes (HBR) depending upon two parameters is similar to the method introduced by Menacer, et al. (2016) for Chua multiscroll attractors. These HBR are characterized by the maximal range extension (MARE) of their attractors and coding the appearance order of the scrolls under the control of the two parameters. Moreover, these HDR have interesting symmetries with respect to the two parameters. The novelty that this article introduces, is firstly the paradigm of MARE and the formula giving their approximate value depending upon parameters p and q, which is linked to the size of the scrolls; secondly the coding of the HBR which is dened for the first time including the basic cell; and thirdly unearthing the symmetries of these routes, allowing to obtain their coding without any numerical computation.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium provided the original work is properly cited.

Highlights

In the last three decades, generation of multiscroll chaotic attractors has been extensively studied due to their promising applications in various real-word technologies
The focus is on the study of the symmetries of hidden bifurcation routes in 1-D multiscroll chaotic attractors generated by saturated function series
The novelty that this article introduces, is rstly the paradigm of the maximal attractor range extension and the formula giving their approxip q mate value depending upon parameters and, which is linked to the size of the scrolls; secondly the coding of the hidden bifurcation routes (HBR) which is dened for the rst time including the basic cell ; and thirdly unearthing the symmetries of these routes, allowing to obtain their coding without any numerical computation

Summary

Introduction

In the last three decades, generation of multiscroll chaotic attractors has been extensively studied due to their promising applications in various real-word technologies. The majority of such multiscroll generations are known for many years, it is only recently that they are studied under the scope of bifurcation theory [2] They have been found for hidden attractors [3] in the case where equilibrium points exist [4], and even in the case of innite number of equilibriums [5]. The focus is on the study of the symmetries of hidden bifurcation routes in 1-D multiscroll chaotic attractors generated by saturated function series. This article is organized as follows: In Section 2, the model of multiscroll chaotic attractors generated by saturated function series proposed in [1] is reviewed. Appendix A presents the analytical-numerical method for hidden attractor localization proposed by Leonov [7]

Multiscrool chaotic attractors from saturated function series

Examples and properties of bifurcation routes

Symmetries of the hidden bifurcation routes

Conclusion

A Analytical-numerical method for hidden attractor localization