Abstract
In this paper a new 5D chaotic system with line equilibrium is designed and described to reveal its extreme multi-stability. Hence, all of the resulting attractors are hidden. The suggested system owns many complex dynamic behaviors in comparison with other chaotic systems. System initial state-associated complex dynamical behaviors are considered and we discover that it possesses an immeasurable number of coexisting attractors, which expresses the occurrence of extreme multi-stability. Besides, we also demonstrate the line equilibrium stability in detail, bifurcation diagrams, Lyapunov exponents, and basins of attraction. Also, in order to analyze the new 5D chaotic system we have considered the permutation entropy technique. Finally, the application of the novel 5D chaotic system with line equilibrium to the problem of chaos synchronization and secure communication through observer design is presented.
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References
W. Ai, K. Sun, Y. Fu, Int. J. Mod. Phys. C 29, 1850049 (2018)
P. Arena, S. Baglio, L. Fortuna, G. Manganaro, Electron. Lett. 31, 250 (1995)
P. Arena, S. Baglio, L. Fortuna, G. Manganaro, , IEEE Trans. Circuits Syst. I: Fundam.Theory Appl. 45, 157 (1998)
P. Arena, L. Fortuna, M. Frasca, L. Patané, IEEE Trans. Neural Networks 20, 202 (2009)
C. Bandt, B. Pompe, Phys. Rev. Lett. 88, 174102 (2002)
B. Bao, H. Wu, L. Xu, M. Chen, W. Hu, Int. J. Bifurc. Chaos 28, 1850019 (2018)
K. Barati, S. Jafari, J.C. Sprott, V.-T. Pham, Int. J. Bifurc. Chaos 26, 1630034 (2016)
M. Boutayeb, M. Darouach, H. Rafaralahy, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 49, 345 (2002)
A. Buscarino, L. Fortuna, M. Frasca, Physica D 238, 1917 (2009)
Y. Cao, W.-W. Tung, J. Gao, V.A. Protopopescu, L.M. Hively, Phys. Rev. E 70, 046217(2004)
R. Caponetto, G. Dongola, L. Fortuna, A. Gallo, Commun. Nonlinear Sci. Numer. Simul. 15, 997 (2010)
S. Chandra, M.K. Gupta, N.K. Tomar, in Int. Conf. Signal Process. Comput. Control (ISPCC) (IEEE, 2015)
U. Chaudhuri, A. Prasad, Phys. Lett. A 378, 713 (2014)
C.-Y. Chen, Z.-H. Guan, M. Chi, Y. Wu, R.-Q. Liao, X.-W. Jiang, J. Franklin Inst. 354, 3120 (2017)
C.-Y. Chen, W.-H. Gui, Z.-H. Guan, R.-L. Wang, S.-W. Zhou, Neurocomputing 226, 101 (2017)
C.-Y. Chen, B. Hu, Z.-H. Guan, M. Chi, D.-X. He, Inf. Sci. 374, 85 (2016)
M. Chen, M. Sun, B. Bao, H. Wu, Q. Xu, J. Wang, Nonlinear Dyn. 91, 1395 (2018)
M. Chen, Q. Xu, Y. Lin, B. Bao, Nonlinear Dyn. 87, 789 (2017)
Z. Faghani, Z. Arab, F. Parastesh, S. Jafari, M. Perc, M. Slavinec, Chaos Solitons Fractals 114, 306 (2018)
N. Fataf, S.K. Palit, S. Mukherjee, M. Said, D.H. Son, S. Banerjee, Eur. Phys. J. Plus 132, 492 (2017)
Y. Feng, J. Pu, Z. Wei, Eur. Phys. J. Special Topics 224, 1593 (2015)
Z.-H. Guan, C.-Y. Chen, G. Feng, T. Li, IEEE Trans. Circuits Syst. I: Regul. Pap. 60, 189(2013)
M.K. Gupta, N.K. Tomar, S. Bhaumik, J. Franklin Inst. 352, 1250 (2015)
M.K. Gupta, N.K. Tomar, V.K. Mishra, S. Bhaumik, Int. J. Appl. Comput. Math. 3, 1313 (2017)
S. He, S. Banerjee, Physica A 490, 366 (2018)
S. He, S. Banerjee, K. Sun, Chaos Solitons Fractals 115, 14 (2018)
S. He, C. Li, K. Sun, S. Jafari, Entropy 20, 556 (2018)
M. Henon, C. Heiles, Astron. J. 69, 73 (1964)
C. Hens, R. Banerjee, U. Feudel, S. Dana, Phys. Rev. E 85, 035202 (2012)
W.G. Hoover, J.C. Sprott, P.K. Patra, Phys. Lett. A 379, 2935 (2015)
B. Hu, Z.-H. Guan, X.-W. Jiang, M. Chi, R.-Q. Liao, C.-Y. Chen, Inf. Sci. 373, 110 (2016)
S. Jafari, A. Ahmadi, A.J.M. Khalaf, H.R. Abdolmohammadi, V.-T. Pham, F.E. Alsaadi, AEU Int. J. Electron. Commun. 89, 131 (2018)
S. Jafari, S. Dehghan, G. Chen, S.T. Kingni, K. Rajagopal, Chaos Solitons Fractals 112, 135 (2018)
S. Jafari, J. Sprott, Chaos Solitons Fractals 57, 79 (2013)
S. Jafari, J. Sprott, S.M.R.H. Golpayegani, Phys. Lett. A 377, 699 (2013)
S. Jafari, J.C. Sprott, Chaos Solitons Fractals 57, 79 (2013)
G.F. Kuiate, K. Rajagopal, S.T. Kingni, V.K. Tamba, S. Jafari, Int. J. Dyn. Control 6, 1008(2018)
N. Kuznetsov, O. Kuznetsova, G. Leonov, V. Vagaitsev, Analytical-numerical Localization of Hidden Attractor in Electrical Chua’s Circuit, in Informatics in Control, Automation and Robotics (Springer, 2013), pp. 149
N. Kuznetsov, V. Vagaytsev, G. Leonov, S. Seledzhi, Localization of hidden attractors in smooth Chua’s systems, in International Conference on Applied and Computational Mathematics (Springer, 2011)
N.V. Kuznetsov, O. Kuznetsova, G.A. Leonov, V. Vagaytsev, Hidden Attractor in Chua’s Circuits, in ICINCO (2011).
G. Leonov, N. Kuznetsov, V. Vagaitsev, Physica D 241, 1482 (2012)
T.-L. Liao, N.-S. Huang, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 46, 1144 (1999)
C. Li, F. Min, Q. Jin, H. Ma, AIP Adv. 7, 125204 (2017)
C. Li, J.C. Sprott, Int. J. Bifurc. Chaos 24, 1450131 (2014)
C. Li, J.C. Sprott, W. Thio, Phys. Lett. A 379, 888 (2015)
C. Li, J.C. Sprott, H. Xing, Phys. Lett. A 380, 1172 (2016)
C. Li, J.C. Sprott, H. Xing, Nonlinear Dyn. 87, 1351 (2017)
M. Molaie, S. Jafari, J.C. Sprott, S.M.R.H. Golpayegani, Int. J. Bifurc. Chaos 23, 1350188 (2013)
M. Molaie, S. Jafari, J.C. Sprott, S.M.R. Hashemi Golpayegani, Int. J. Bifurc. Chaos 23, 1350188 (2013)
L. Moysis, C. Volos, V.-T. Pham, S. Goudos, I. Stouboulos, M.K. Gupta, Synchronization of a chaotic system with line equilibrium using a descriptor observer for secure communicationin 2019 8th International Conference on Modern Circuits and Systems Technologies (MOCAST) (IEEE, 2019), pp. 1–4
T. Ogawa, Phys. Rev. A 37, 4286 (1988)
Y. Peng, K. Sun, S. He, X. Yang, Eur. Phys. J. Plus 133, 305 (2018)
V.-T. Pham, S. Jafari, C. Volos, X. Wang, S.M.R.H. Golpayegani, Int. J. Bifurc. Chaos 24, 1450146 (2014)
K. Rajagopal, S. Çiçek, A.J.M. Khalaf, V.-T. Pham, S. Jafari, A. Karthikeyan, P. Duraisamy, Z. Naturforsch. A 73, 609 (2018)
X. Ren, P. Ren, B. Chen, J.C. Principe, N. Zheng, A reconfigurable parallel acceleration platform for evaluation of permutation entropy, in 2014 36th Annual International Conference of the IEEE Engineering in Medicine and Biology Society, (IEEE, 2014)
M. Riedl, A. Müller, N. Wessel, Eur. Phys. J. Special Topics 222, 249 (2013)
J. Ruan, K. Sun, J. Mou, S. He, L. Zhang, Eur. Phys. J. Plus 133, 3 (2018)
P.S. Shabestari, S. Panahi, B. Hatef, S. Jafari, J.C. Sprott, Chaos Solitons Fractals 112, 44 (2018)
X. Sun, M. Perc, J. Kurths, Chaos 27, 053113 (2017)
A. Vicari, A. Ciraudo, C. Del Negro, A. Herault, L. Fortuna, Nat. Hazard 50, 539 (2009)
H. Wang, X.-J. Zhu, S.-W. Gao, Z.-Y. Chen, Commun. Nonlinear Sci. Numer. Simul. 16, 1517 (2011)
H. Wang, Z. Han, W. Zhang, Q. Xie, Nonlinear Dyn. 57, 69 (2009)
G. Wang, C. Shi, X. Wang, F. Yuan, Math. Prob. Eng. 2017, 6504969 (2017)
R. Wang, K. Mei, C. Chen, Y. Li, H. Mei, Z. Yu, J. Control Theory Appl. 10, 309 (2012)
X. Wang, G. Chen, Commun. Nonlinear Sci. Numer. Simul. 17, 1264 (2012)
Z. Wang, H.R. Abdolmohammadi, F.E. Alsaadi, T. Hayat, V.-T. Pham, Chaos Solitons Fractals 110, 252 (2018)
Z. Wei, I. Moroz, J. Sprott, A. Akgul, W. Zhang, Chaos 27, 033101 (2017)
Z. Wei, I. Moroz, J.C. Sprott, Z. Wang, W. Zhang, Int. J. Bifurc. Chaos 27, 1730008 (2017)
Z. Wei, I. Moroz, Z. Wang, J.C. Sprott, T. Kapitaniak, Int. J. Bifurc. Chaos 26, 1650125 (2016)
Z. Wei, V.-T. Pham, T. Kapitaniak, Z. Wang, Nonlinear Dyn. 85, 1635 (2016)
Z. Wei, Y. Tang, H. Chen, I. Pehlivan, Optoelectron. Adv. Mater. Rapid Commun. 7, 984 (2013)
Z. Wei, R. Wang, A. Liu, Math. Comput. Simul. 100, 13 (2014)
Z. Wei, P. Yu, W. Zhang, M. Yao, Nonlinear Dyn. 82, 131 (2015)
Z. Wei, W. Zhang, Int. J. Bifurc. Chaos 24 (2014)
Y.-X. Tang, A.J.M.K., K. Rajagopal, V.-T. Pham, S. Jafari, Y. Tian, Chin. Phys. B 27, 40502 (2018)
L.-M. Zhang, K.-H. Sun, W.-H. Liu, S.-B. He, Chin. Phys. B 26, 100504 (2017)
X.-W. Zhao, B. Hu, Z.-H. Guan, C.-Y. Chen, M. Chi, X.-H. Zhang, IET Control Theory Appl. 10, 2093 (2016)
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Khalaf, A.J.M., Abdolmohammadi, H.R., Ahmadi, A. et al. Extreme multi-stability analysis of a novel 5D chaotic system with hidden attractors, line equilibrium, permutation entropy and its secure communication scheme. Eur. Phys. J. Spec. Top. 229, 1175–1188 (2020). https://doi.org/10.1140/epjst/e2020-900225-9
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DOI: https://doi.org/10.1140/epjst/e2020-900225-9