×
Mahmoud, Gamal M.; Al-Kashif, M. A.; Aly, Shaban A.

Basic properties and chaotic synchronization of complex Lorenz system. (English) Zbl 1115.37035

Int. J. Mod. Phys. C 18, No. 2, 253-265 (2007).
Summary: This paper aims at studying the basic properties and chaotic synchronization of the complex Lorenz system: \[ \dot x=\alpha(y-x), \quad\dot y=\gamma x-y-xz,\quad \dot z=-\beta z+\frac 12(\overline xy+x\overline y),\tag{*} \] where \(\alpha, \gamma,\beta\) are positive (real or complex) parameters, \(x\) and \(y\) are complex variables, \(z\) is a real variable, an overbar denotes complex conjugate variable and dots represent derivatives with respect to time. This system arises in many important applications in physics, for example, in laser physics and rotating fluids dynamics. Numerically we show that this system is a chaotic system and exhibits chaotic attractors. The necessary conditions for system (*) to generate chaos are obtained. Analytical and numerical calculations are presented to achieve synchronization. Active control technique is used to synchronize chaotic attractors of equations (*).

Cited in 43 Documents

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
93C10 Nonlinear systems in control theory
93D15 Stabilization of systems by feedback

Keywords:

chaotic attractor; chaos; active control; error system; equilibria; dissipation
PDFBibTeX XMLCite
\textit{G. M. Mahmoud} et al., Int. J. Mod. Phys. C 18, No. 2, 253--265 (2007; Zbl 1115.37035)
Full Text: DOI

References:

[1] DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 · Zbl 1417.37129 · doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
[2] DOI: 10.1016/0375-9601(78)90196-2 · doi:10.1016/0375-9601(78)90196-2
[3] DOI: 10.1007/BF01011469 · doi:10.1007/BF01011469
[4] Tritton D. J., Physical Fluid Dynamics (1988) · Zbl 0383.76001
[5] Ning C. Z., Phys. Rev. A 41 pp 3827–
[6] DOI: 10.1016/j.chaos.2004.11.032 · Zbl 1198.34069 · doi:10.1016/j.chaos.2004.11.032
[7] DOI: 10.1016/0375-9601(83)90052-X · doi:10.1016/0375-9601(83)90052-X
[8] DOI: 10.1016/S0960-0779(00)00216-2 · Zbl 1073.93537 · doi:10.1016/S0960-0779(00)00216-2
[9] DOI: 10.1016/S0960-0779(97)00161-6 · Zbl 1047.37502 · doi:10.1016/S0960-0779(97)00161-6
[10] DOI: 10.1016/S0016-0032(99)00010-1 · Zbl 1051.93514 · doi:10.1016/S0016-0032(99)00010-1
[11] DOI: 10.1007/978-1-4612-5767-7 · doi:10.1007/978-1-4612-5767-7
[12] DOI: 10.1016/S0960-0779(98)00328-2 · Zbl 0985.37106 · doi:10.1016/S0960-0779(98)00328-2
[13] DOI: 10.1016/S0960-0779(04)00432-1 · doi:10.1016/S0960-0779(04)00432-1
[14] DOI: 10.1016/S0167-2789(96)00129-7 · Zbl 0887.34048 · doi:10.1016/S0167-2789(96)00129-7
[15] DOI: 10.1016/0167-2789(85)90176-9 · Zbl 0579.76051 · doi:10.1016/0167-2789(85)90176-9
[16] DOI: 10.1016/0167-2789(84)90441-X · Zbl 0593.58025 · doi:10.1016/0167-2789(84)90441-X
[17] Roberts P. H., Rev. Mod. Phys. 72 pp 1083–
[18] Vladimirov A. G., Int. J. Bifurcation and Chaos 18 pp 723–
[19] DOI: 10.1016/0167-2789(82)90053-7 · Zbl 1194.76280 · doi:10.1016/0167-2789(82)90053-7
[20] Panchev S., J. Calcutta Math. Soc. 1 pp 181–
[21] George P., J. Phys. A 22 pp 137–
[22] DOI: 10.1103/PhysRevE.55.3689 · doi:10.1103/PhysRevE.55.3689
[23] DOI: 10.1142/S0218127404011740 · Zbl 1090.37516 · doi:10.1142/S0218127404011740
[24] DOI: 10.1016/j.chaos.2004.11.031 · Zbl 1125.93469 · doi:10.1016/j.chaos.2004.11.031
[25] DOI: 10.1016/j.chaos.2004.11.042 · Zbl 1125.93473 · doi:10.1016/j.chaos.2004.11.042
[26] DOI: 10.1016/j.physleta.2003.11.027 · Zbl 1065.93028 · doi:10.1016/j.physleta.2003.11.027
[27] DOI: 10.1016/S0375-9601(03)00573-5 · Zbl 1024.37053 · doi:10.1016/S0375-9601(03)00573-5
[28] Wu X., Chaos, Solitons & Fractals 18 pp 721–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.