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Aly, Shaban; Al-Qahtani, Ali; Khenous, Houari B.; Mahmoud, Gamal M.

Impulsive control and synchronization of complex Lorenz systems. (English) Zbl 1474.34361

Abstr. Appl. Anal. 2014, Article ID 932327, 9 p. (2014).
Summary: In this paper, we continue our investigations on control and synchronization of the complex Lorenz systems by investigating impulsive control and synchronization. Nonlinear systems involving impulse effects, appear as a natural description of observed evolution phenomena of several real world problems; For example, many biological phenomena involving thresholds, bursting rhythm models in medicine and biology, optimal control models in economics, population dynamics and so forth do exhibit impulsive effects. Some new and more comprehensive criteria for global exponential stability and asymptotical stability of impulsively controlled complex Lorenz systems are established with varying impulsive intervals. The effectiveness of the proposed technique is verified through numerical simulations.

Cited in 2 Documents

MSC:

34D06 Synchronization of solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34H10 Chaos control for problems involving ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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\textit{S. Aly} et al., Abstr. Appl. Anal. 2014, Article ID 932327, 9 p. (2014; Zbl 1474.34361)
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References:

[1] Mahmoud, G. M.; Aly, S. A.; AL-Kashif, M. A., Dynamical properties and chaos synchronization of a new chaotic complex nonlinear system, Nonlinear Dynamics, 51, 1-2, 171-181 (2008) · Zbl 1170.70365
[2] Mahmoud, G. M.; Al-Kashif, M. A.; Aly, S. A., Basic properties and chaotic synchronization of complex Lorenz system, International Journal of Modern Physics C: Computational Physics, Physical Computation, 18, 2, 253-265 (2007) · Zbl 1115.37035
[3] Mahmoud, G. M.; Bountis, T., The dynamics of systems of complex nonlinear oscillators: a review, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 14, 11, 3821-3846 (2004) · Zbl 1091.34524
[4] Mahmoud, G. M., Approximate solutions of a class of complex nonlinear dynamical systems, Physica A: Statistical Mechanics and its Applications, 253, 1-4, 211-222 (1998)
[5] Schmutz, M.; Rueff, M., Bifurcation schemes of the Lorenz model, Physica D: Nonlinear Phenomena, 11, 1-2, 167-178 (1984) · Zbl 0593.58025
[6] Fowler, A. C.; Gibbon, J. D.; McGuinness, M. J., The real and complex Lorenz equations and their relevance to physical systems, Physica D: Nonlinear Phenomena, 7, 1-3, 126-134 (1983) · Zbl 1194.76087
[7] Ning, C.-Z.; Haken, H., Detuned lasers and the complex Lorenz equations: subcritical and supercritical Hopf bifurcations, Physical Review A, 41, 7, 3826-3837 (1990)
[8] Vladimirov, A. G.; Toronov, V. Yu.; Derbov, V. L., The complex Lorenz model: geometric structure, homoclinic bifurcation and one-dimensional map, International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, 8, 4, 723-729 (1998) · Zbl 1109.37303
[9] Jones, C. A.; Weiss, N. O.; Cattaneo, F., Nonlinear dynamos: a complex generalization of the Lorenz equations, Physica D: Nonlinear Phenomena, 14, 2, 161-176 (1985) · Zbl 0579.76051
[10] Flessas, G. P., New exact solutions of the complex Lorenz equations, Journal of Physics A: Mathematical and General, 22, 5, L137-L141 (1989) · Zbl 0682.70006
[11] Roberts, P. H.; Glazmaier, G. A., Geodynamo theory and simulations, Reviews of Modern Physics, 72, 1083-1123 (2000)
[12] Panchev, S.; Vitanov, N. K., On asymptotic properties of some complex Lorenz-like systems, Journal of the Calcutta Mathematical Society, 1, 3-4, 181-190 (2005) · Zbl 1105.34032
[13] Toronov, V. Yu.; Derbov, V. L., Boundedness of attractors in the complex Lorenz model, Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, 55, 3, 3689-3692 (1997)
[14] Tritton, D. J., Physical Fluid Dynamics. Physical Fluid Dynamics, Oxford Science Publications (1988), New York, NY, USA: The Clarendon Press, Oxford University Press, New York, NY, USA · Zbl 0383.76001
[15] Rauh, A.; Hannibal, L.; Abraham, N. B., Global stability properties of the complex Lorenz model, Physica D: Nonlinear Phenomena, 99, 1, 45-58 (1996) · Zbl 0887.34048
[16] Yorke, J. A.; Yorke, E. D., Metastable chaos: the transition to sustained chaotic behavior in the Lorenz model, Journal of Statistical Physics, 21, 3, 263-277 (1979)
[17] Shimizu, T.; Morioka, N., Transient behavior in periodic regions of the Lorenz model, Physics Letters A, 69, 3, 148-150 (1978)
[18] Pecora, L. M.; Carroll, T. L., Synchronization in chaotic systems, Physical Review Letters, 64, 8, 821-824 (1990) · Zbl 0938.37019
[19] Mahmoud, G. M.; Rauh, A.; Mohamed, A. A., On modulated complex non-linear dynamical systems, Il Nuovo Cimento della Società Italiana di Fisica B: Serie 12, 114, 1, 31-47 (1999)
[20] Mahmoud, G. M.; Rauh, A.; Farghaly, A. A. M., Applying chaos control to a modulated complex nonlinear system, Nuovo Cimento della Societa Italiana di Fisica B, 116, 10, 113-126 (2001)
[21] Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Physical Review Letters, 64, 11, 1196-1199 (1990) · Zbl 0964.37501
[22] Lu, J.; Wu, X.; Lü, J., Synchronization of a unified chaotic system and the application in secure communication, Physics Letters A, 305, 6, 365-370 (2002) · Zbl 1005.37012
[23] Mahmoud, G. M.; Aly, S. A., Periodic attractors of complex damped non-linear systems, International Journal of Non-Linear Mechanics, 35, 2, 309-323 (2000) · Zbl 1068.70525
[24] Mahmoud, G. M.; Aly, S. A.; Farghaly, A. A., On chaos synchronization of a complex two coupled dynamos system, Chaos, Solitons and Fractals, 33, 1, 178-187 (2007) · Zbl 1152.37317
[25] Liao, T.-L., Adaptive synchronization of two Lorenz systems, Chaos, Solitons and Fractals, 9, 9, 1555-1561 (1998) · Zbl 1047.37502
[26] Liao, T.-L.; Lin, S.-H., Adaptive control and synchronization of Lorenz systems, Journal of the Franklin Institute: Engineering and Applied Mathematics, 336, 6, 925-937 (1999) · Zbl 1051.93514
[27] Hu, J.; Wang, Z.; Shen, B.; Gao, H., Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements, International Journal of Control, 86, 4, 650-663 (2013) · Zbl 1278.93269
[28] Hu, J.; Chen, D.; Du, J., State estimation for a class of discrete nonlinear systems with randomly occurring uncertainties and distributed sensor delays, International Journal of General Systems, 43, 3-4, 387-401 (2014) · Zbl 1302.93201
[29] Hu, J.; Wang, Z.; Gao, H., Recursive filtering with random parameter matrices, multiple fading measurements and correlated noises, Automatica, 49, 11, 3440-3448 (2013) · Zbl 1315.93081
[30] Richter, H., Controlling the Lorenz system: combining global and local schemes, Chaos, Solitons and Fractals, 12, 13, 2375-2380 (2001) · Zbl 1073.93537
[31] Stewart, I., The Lorenz attractor exists, Nature, 406, 948-949 (2000)
[32] Lakshmikantham, V.; Baĭnov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations. Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics, 6 (1989), Singapore: World Scientific Publishing, Singapore · Zbl 0719.34002
[33] Yang, T.; Yang, L.-B.; Yang, C.-M., Impulsive control of Lorenz system, Physica D: Nonlinear Phenomena, 110, 1-2, 18-24 (1997) · Zbl 0925.93414
[34] Chen, S.; Yang, Q.; Wang, C., Impulsive control and synchronization of unified chaotic system, Chaos, Solitons and Fractals, 20, 4, 751-758 (2004) · Zbl 1050.93051
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