The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particularly through structures of bifurcations or chaotic synchronization. A bifurcation is a qualitative change of the system behavior under the influence of control parameters. This change may correspond to the disppearance or appearance of new singularities or a change in the nature of singularities. We can define different kinds of bifurcations for fixed points and period two cycles as, saddle-node, period doubling, transcritical or pitchfork bifurcations. The study of the sequence of bifurcations permits to understand the mechanisms that lead to chaos. The phenomena of synchronization and antisynchronization in coupled chaotic systems is very important because its applications in several areas, such as secure communication or biology. In this paper, we study bifurcation properties of a two-dimensional coupled map T with three parameters. The first objective is to locate the bifurcation curves and their evolution in the parametric plane (a,b), when a third parameter c varies. The equations of some bifurcation curves are given analytically; cusp points and co-dimension two points on these bifurcation curves are determined. The second is related to the study of the chaotic synchronization and antisynchronization in the phase space (x,y).
| Published in | Applied and Computational Mathematics (Volume 11, Issue 1) |
| DOI | 10.11648/j.acm.20221101.12 |
| Page(s) | 18-30 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Two Dimensional Coupled Map, Bifurcation, Chaotic Synchronization and Antisynchronization
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APA Style
Yamina Soula, Abdel Kaddous Taha, Daniele Fournier-Prunaret, Nasr-Eddine Hamri. (2022). Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps. Applied and Computational Mathematics, 11(1), 18-30. https://doi.org/10.11648/j.acm.20221101.12
ACS Style
Yamina Soula; Abdel Kaddous Taha; Daniele Fournier-Prunaret; Nasr-Eddine Hamri. Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps. Appl. Comput. Math. 2022, 11(1), 18-30. doi: 10.11648/j.acm.20221101.12
AMA Style
Yamina Soula, Abdel Kaddous Taha, Daniele Fournier-Prunaret, Nasr-Eddine Hamri. Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps. Appl Comput Math. 2022;11(1):18-30. doi: 10.11648/j.acm.20221101.12
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Download@article{10.11648/j.acm.20221101.12,
author = {Yamina Soula and Abdel Kaddous Taha and Daniele Fournier-Prunaret and Nasr-Eddine Hamri},
title = {Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps},
journal = {Applied and Computational Mathematics},
volume = {11},
number = {1},
pages = {18-30},
doi = {10.11648/j.acm.20221101.12},
url = {https://doi.org/10.11648/j.acm.20221101.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20221101.12},
abstract = {The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particularly through structures of bifurcations or chaotic synchronization. A bifurcation is a qualitative change of the system behavior under the influence of control parameters. This change may correspond to the disppearance or appearance of new singularities or a change in the nature of singularities. We can define different kinds of bifurcations for fixed points and period two cycles as, saddle-node, period doubling, transcritical or pitchfork bifurcations. The study of the sequence of bifurcations permits to understand the mechanisms that lead to chaos. The phenomena of synchronization and antisynchronization in coupled chaotic systems is very important because its applications in several areas, such as secure communication or biology. In this paper, we study bifurcation properties of a two-dimensional coupled map T with three parameters. The first objective is to locate the bifurcation curves and their evolution in the parametric plane (a,b), when a third parameter c varies. The equations of some bifurcation curves are given analytically; cusp points and co-dimension two points on these bifurcation curves are determined. The second is related to the study of the chaotic synchronization and antisynchronization in the phase space (x,y).},
year = {2022}
}
TY - JOUR T1 - Bifurcations and Dynamical Behavior of 2D Coupled Chaotic Sine Maps AU - Yamina Soula AU - Abdel Kaddous Taha AU - Daniele Fournier-Prunaret AU - Nasr-Eddine Hamri Y1 - 2022/02/18 PY - 2022 N1 - https://doi.org/10.11648/j.acm.20221101.12 DO - 10.11648/j.acm.20221101.12 T2 - Applied and Computational Mathematics JF - Applied and Computational Mathematics JO - Applied and Computational Mathematics SP - 18 EP - 30 PB - Science Publishing Group SN - 2328-5613 UR - https://doi.org/10.11648/j.acm.20221101.12 AB - The main characteristics of a dynamical system are determined by the bifurcation theory. In particular, in this paper we examined the properties of the discrete dynamical system of a two coupled maps, i.e. the maps with an invariant unidimensional submanifold. The study of coupled chaotic systems shows rich and complex dynamic behaviors, particularly through structures of bifurcations or chaotic synchronization. A bifurcation is a qualitative change of the system behavior under the influence of control parameters. This change may correspond to the disppearance or appearance of new singularities or a change in the nature of singularities. We can define different kinds of bifurcations for fixed points and period two cycles as, saddle-node, period doubling, transcritical or pitchfork bifurcations. The study of the sequence of bifurcations permits to understand the mechanisms that lead to chaos. The phenomena of synchronization and antisynchronization in coupled chaotic systems is very important because its applications in several areas, such as secure communication or biology. In this paper, we study bifurcation properties of a two-dimensional coupled map T with three parameters. The first objective is to locate the bifurcation curves and their evolution in the parametric plane (a,b), when a third parameter c varies. The equations of some bifurcation curves are given analytically; cusp points and co-dimension two points on these bifurcation curves are determined. The second is related to the study of the chaotic synchronization and antisynchronization in the phase space (x,y). VL - 11 IS - 1 ER -
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