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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Two Dimensional Coupled Map, Bifurcation, Chaotic Synchronization and Antisynchronization
| [1] | J. C. Alexander, J. A. Yorke, Z. You, and I. Kan, (1992), “Riddled basins,” Int. J. Bifurc. Chaos 2, 795-813. |
| [2] | J. C. Alexander, B. R. Hunt, I. Kan, and J. A.Yorke, (1996) “Intermingled basins of the triangle map,” Ergod. Th. Dyn. Syst. 16, 651-662. |
| [3] | J.P. Carcasses, (1993), “Determination of different configurations of fold and flip bifurcation curves of a one or two-dimensional map,” Int. J. Bifurc. Chaos Vol. 3, No 4, 869-902. |
| [4] | J.P. Carcasses, (1995) “Singularities of the parametric plane of an n-dimensional map. Determination of different configurations of fold and flip bifurcation curves,” Int. J. Bifurc. Chaos Vol. 5, No 2, 419-447. |
| [5] | K. M. Cuomo and A. V. Oppenheim, (1993) “Circuit implementation of synchronized chaos with applications to communications,” Phys. Rev. Lett. 71, 65-68. |
| [6] | S. Dinar, (1994) “Etude des bifurcations d’une equation non autonome de Duffing-Rayleigh et d’un modulateur MICDIF,” PhD These, Universite Paul Sabatier de Toulouse, No 1673. |
| [7] | H. Fujisaka and T. Yamada, (1983) “Stability theory of synchronized motion in coupled-oscillator systems,” Progr. Theor. Phys. 69, 32-46. |
| [8] | L. Gardini, R. Abraham, R. J. Record and D. Fournier- Prunaret, (1994) “A double logistic map,” Int. J. Bifurc. Chaos, Vol. 4, No. 1, 145-176. |
| [9] | I. Gumowski and C. Mira, (1980) “Dynamique chaotique, transformations ponctuelles, transition ordre desordre,” Ed. Ce padues. |
| [10] | N. E. Hamri, Y. Soula, (2010) “Basins and critical curves generated by a family of two-dimensional sine maps,” Electronic Journal of Theoretical Physics, EJTP 7, No. 24, 139-150. |
| [11] | L. Kocarev, K. S. Halle, K. Eckert, L. O. Chua, and U. Parlity, (1992) “Transmission of Digital signals by chaotic synchronization,” Int. J. Bifurc. Chaos, Appl. Sci. Eng. 2, 973-977. |
| [12] | V. L. Maistrenko, YU. L. Maistrenko and E. Mosekilde, (2005), “Chaotic synchronization and antisynchronization in coupled sine maps,” Int. J. Bifurc. Chaos Vol. 15, No 7, 2161-2177. |
| [13] | Yu. L. Maistrenko, V. L. Maistrenko, A. Popovich and E. Mosekilde, (1998), “Transverse instability and riddled basins in a system of two coupled logistic maps,” Phys. Rev. E57, 2713–2724. |
| [14] | J. Leonel Rocha, Abdel-Kaddous Taha and D. Fournier- Prunaret, (2016), “Big Bang Bifurcation Analysis and Allee Effect in Generic Growth Functions,” International Journal of Bifurcation and Chaos, Vol. 26, No. 6 1650108, (20 pages), World Scientific Publishing Company, DOI: 10.1142/S021812741650108 X. |
| [15] | J. Leonel Rocha, Abdel-Kaddous Taha and D. Fournier-Prunaret, (2016), “Dynamical Analysis and Big Bang Bifurcations of 1D and 2D Gompertz’s Growth Functions,” International Journal of Bifurcation and Chaos, Vol. 26 , No 11 1630030, (22 pages), World Scientific Publishing Company, DOI: 10.1142/S0218127416300305. |
| [16] | J. Leonel Rocha and Abdel-Kaddous Taha, (2019), “Allee’s Effect Bifurcation in Generalized Logistic Maps”, International Journal of Bifurcation and ChaosVol. 29 , No. 03,1950039. |
| [17] | J. Leonel Rocha and Abdel-Kaddous Taha, (2019), “Bifurcation structures in a 2D exponential diffeomorphismwith Allee effect,” Nonlinear Dyn 95:3357-3374, https://doi.org/10.1007/s11071-019- 04759-3. |
| [18] | J. Leonel Rocha, Abdel-Kaddous Taha and D. Fournier- Prunaret, (2020), “Dynamics and bifurcations of a map of homographic Ricker type,” Nonlinear Dyn 102: 1129- 1149 https://doi.org/ 10.1007 /s 11071-020-05820-2. |
| [19] | L. Gardini, A, Garijo and X. Jarque, (2021), “Topological Properties of the Immediate Basins of Attraction for the Secant Method,” Mediterr. J. Math. 18:221 https://doi.org/10.1007/s00009-021- 01845-y1660-5446/21/050001-27 published online September 7, 2021 The Author(s) 2021. |
| [20] | C. Mira, and C. Gracio, (2003), “On the embedding of a (p-1)-dimensional noninvertible map into a pdimensional invertible map (p=2, 3),” Int. J. Bifurcation and Chaos 13, 1787-1810. |
| [21] | C. Mira, and L. Gardini, (2009), “From the box-withina- box bifurcation organization to the Julia set. Part I: Revisited properties of the sets generated by a quadratic complex map with a real parameter,” Int. J. Bifurcation and Chaos 19, 281–327. |
| [22] | C. Mira, (2013), “Embedding of a Dim1 piecewise continuous and linear Leonov map into a Dim2 invertible map,” “Global Analysis of Dynamic Models in Economics and Finance,” eds. G. I. Bischi, et al. (Springer, Berlin, Heidelberg), pp. 337-367. |
| [23] | H. Li, K. Li, M. Chen and B. Bao, (2020), “Coexisting infinite orbits in an area-preserving Lozi map,” Entropy 22, 1119. |
| [24] | F. Tramontana and L. Gardini, (2021), “Revisiting Samuelson’s models, linear and nonlinear, stability conditions and oscillating dynamics,” Tramontana and Gardini Economic Structures 10:9 https://doi.org/10.1186/s40008-021-00239-3, Journal of Economic Structures. |
| [25] | M. Misiurewicz, and S. Stimac, (2016), “Symbolic dynamics for Lozi maps,” Nonlinearity 29, 3031 |
| [26] | D.Simpson, (2014), “Sequencesofperiodicsolutionsand infinitelymanycoexistingattractorsinthebordercollision normal form,” Int. J. Bifurcation and Chaos 24, 1430018- 1-18. |
| [27] | I. Sushko V. Avrutin and L. Gardini, (2021), “Center Bifurcation in the Lozi Map,” International Journal of Bifurcation and Chaos, Vol. 31, No. 16 2130046 (28 pages), World Scientific Publishing Company DOI: 10.1142/S0218127421300469. |
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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