Abstract
Hidden attractors in chaotic dynamical systems can be found by exploring the basin of attraction which has no intersect with any equilibria. Controlling chaos in these systems are complicated, which needs developed methods. In this paper, a 3D jerk system with only one stable equilibrium and hidden attractor is analyzed in infinity by the help of the Poincare compactification in R3. Meanwhile, a distributed delayed feedback (DDF) control scheme for this system is proposed. By using the center manifold theory of functional differential equation (FDE), Hopf bifurcation for the DDF control system is analyzed and obtained. Results confirm the accuracy of the bifurcation analysis and the effectiveness of the proposed DDF control strategy.
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References
S. Jafari, J.C. Sprott, S. Mohammad, Phys. Lett. A 377, 699 (2013)
Z.C. Wei, Phys. Lett. A 376, 102 (2011)
V.T. Pham, C. Volos, S., Jafari, A. Ouannas, T.T. Dao, in 2019 18th European Control Conference (ECC) (2019), p. 2603
X. Wang, G.R. Chen, Nonlinear Dyn. 71, 429 (2013)
S. Panahi, V.T. Pham, K. Rajagopal, O. Boubaker, S. Jafari, in Recent advances in chaotic systems and synchronization (Academic Press, 2019), pp. 63–76
S. Jafari, J.C. Sprott, Chaos Solitons Fractals 57, 79 (2013)
A. Ahmadi, K. Rajagopal, F.E. Alsaadi, V.T. Pham, F.E. Alsaadi, S. Jafari, IJST-T Electron. Eng. 44, 59 (2020)
E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963)
G.R. Chen, T. Ueta, Int. J. Bifurc. Chaos 9, 1465 (1999)
J.H. Lü, G.R. Chen, Int. J. Bifurc. Chaos 12, 659 (2002)
K. Rajagopal, F. Nazarimehr, S. Jafari, A. Karthikeyan, Eur. Phys. J. Special Topics 226, 3827 (2017)
Z. Wang, C. Volos, S.T. Kingni, A.T. Azar, V.T. Pham, Optik 131, 1071 (2017)
Z. Wang, I. Moroz, Z. Wei, H.P. Ren, Pramana 90, 12 (2018)
D. Dudkowski, S. Jafari, T. Kapitaniak, N.V. Kuznetsov, G.A. Leonov, A. Prasad, Phys. Rep. 637, 1 (2016)
H.R. Abdolmohammadi, A.J.M. Khalaf, S. Panahi, K. Rajagopal, V.T. Pham, S. Jafari, Pramana 90, 70 (2018)
G.A. Leonov, N.V. Kuznetsov, Adv. Intell. Syst. Comput. 210, 5 (2013)
Ü. Çavusoğlu, S. Panahi, A. Akgül, S. Jafari, S. Kaçar, Analog Integr. Circ. Sig. Process 98, 85 (2019)
M. Nourian Zavareh, F. Nazarimehr, K. Rajagopal, S. Jafari, J. Int. Bifurc. Chaos 28, 1850171 (2018)
N.V. Kuznetsov, G.A. Leonov, V.I. Vagaitsev, IFAC Proc. 4, 29 (2010)
G.A. Leonov, N.V. Kuznetsov, IFAC Proc. 44, 2494 (2011)
E. Ott, C. Grebogi, J.A. York, Phys. Rev. Lett. 64, 1196 (1990)
Z. Wang, W. Sun, Z.C. Wei, X.J. Xi, Kybernetika 50, 616 (2014)
Z. Wang, Control Theory Appl. 28, 1306 (2011)
I. Karafyllis, in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (2005), p. 5776
J.B. Guan, F.Y. Chen, G.X. Wang, Adv. Differ. Equ. 2012, 166 (2012)
C.J. Xu, Y.S. Wu, Int. J. Bifurc. Chaos 12, 182 (2015)
C.U. Choe, R.S. Kim, H. Jang, P. Hovel, E. Scholl, Int. J. Dyn. Control 2, 2 (2014)
Z. Wang, Nonlinear Dyn. 82, 577 (2015)
Z. Wang, W. Sun, Z.C. Wei, S.W. Zhang, Kybernetika 53, 354 (2017)
J. Hale, Theory of functional differential equations (Springer, New York, 1977)
S.G. Ruan, J.J. Wei, J. Math. Appl. Med. Biol. 18, 41 (2001)
S.G. Ruan, J.J. Wei, Dyn. Continu. Discret. Impul. Syst. 10, 863 (2003)
B. Hassard, N. Kazarinoff, Y. Wan, Theory and applications of Hopf bifurcation (Cambridge University Press, Cambridge, 1981)
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Wang, Z., Xi, X., Kong, L. et al. Infinity dynamics and DDF control for a chaotic system with one stable equilibrium. Eur. Phys. J. Spec. Top. 229, 1319–1333 (2020). https://doi.org/10.1140/epjst/e2020-900134-4
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DOI: https://doi.org/10.1140/epjst/e2020-900134-4