Abstract.
The aim of this paper is to introduce a generalized form of the Lorenz system with time delay. Instead of considering each state variable of the Lorenz system belonging to \(\mathbb{R}\), the paper considers two of them belonging to \(\mathbb{R}^{n}\). Hence the Lorenz system has (2n+1) dimension. This system appears in several applied sciences such as engineering, physics and networks. The stability of the trivial and nontrivial fixed points and the existence of Hopf bifurcations are studied analytically. Using the normal form theory and center manifold argument, the direction and the stability of the bifurcating periodic solutions are determined. Finally, numerical simulations are calculated to confirm our theoretical results. The paper concludes that the dynamics of this system are rich. Additionally, the values of the delay parameter at which chaotic and hyperchaotic solutions exist for different values of n using Lyapunov exponents and Kolmogorov-Sinai entropy are calculated numerically.
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References
E.N. Lorenz, J. Atmos. Sci. 20, 130 (1963)
G. Chen, T. Ueta, Int. J. Bifurc. Chaos 9, 1465 (1999)
O.E. Rössler, Phys. Lett. A 57, 397 (1976)
J. Lü, G. Chen, S. Zhang, Chaos Solitons Fractals 14, 669 (2002)
L.O. Chua, M. Komuro, T. Matsumoto, IEEE Trans. Circuits Syst. 33, 1072 (1986)
A.C. Fowler, M.J. McGuinness, J.D. Gibbon, Physica D 4, 139 (1982)
G.M. Mahmoud, M.A. Al-Kashif, S.A. Aly, Int. J. Mod. Phys. C 18, 253 (2007)
L. Cveticanin, Mech. Mach. Theory 30, 581 (1995)
C.Z. Ning, H. Haken, Phys. Rev. A 41, 3826 (1990)
A.C. Newell, J.V. Moloney, Nonlinear Optics (Addison-Wesley, 1992)
H. Richter, Chaos Solitons Fractals 12, 2375 (2001)
S.I. Niculescu, Delay Effects on Stability: A Robust Control Approach (Springer, Heidelberg, 2001)
N. MacDonald, Biological Delay Systems: Linear Stability Theory (Cambridge University Press, Cambridge, 1989)
Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic Press, Boston, 1993)
M.C. Mackey, L. Glass, Science 197, 287 (1977)
K. Ikeda, Opt. Commun. 30, 257 (1979)
D. Ghosh, R. Chowdhury, P. Saha, Chaos Solitons Fractals 35, 472 (2008)
A. Sharma, M.D. Shrimali, A. Prasad, R. Ramaswamy, Eur. Phys. J. ST 226, 1903 (2017)
T. Banerjee, D. Biswas, B.C. Sarkar, Nonlinear Dyn. 70, 721 (2012)
G.M. Mahmoud, E.E. Mahmoud, A.A. Arafa, Nonlinear Dyn. 80, 855 (2015)
R. Jaimes-Reátegui et al., Eur. Phys. J. ST 225, 2707 (2016)
Z. Wang, W. Sun, Z. Wei, S. Zhang, Nonlinear Dyn. 82, 577 (2015)
V.-T. Pham et al., Eur. Phys. J. ST 225, 127 (2016)
B. Hassard, N. Kazarinoff, Y. Wan, Theory and Applications of Hopf Bifurcation (Cambridge University Press, Cambridge, 1981)
Y. Song, J. Wei, Chaos Solitons Fractals 22, 75 (2004)
D. Breda, E. Van Vleck, Numer. Math. 126, 225 (2013)
R. Vicente, J. Daudén, P. Colet, R. Toral, IEEE J. Quantum Elect. 41, 541 (2005)
A. Uchida, Optical Communication with Chaotic Lasers: Applications of Nonlinear Dynamics and Synchronization (John Wiley & Sons, 2012)
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Mahmoud, G.M., Arafa, A.A. & Mahmoud, E.E. Bifurcations and chaos of time delay Lorenz system with dimension 2n+1. Eur. Phys. J. Plus 132, 461 (2017). https://doi.org/10.1140/epjp/i2017-11739-6
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DOI: https://doi.org/10.1140/epjp/i2017-11739-6