DYNAMICAL BEHAVIOR OF THE GENERALIZED COMPLEX LORENZ CHAOTIC SYSTEM

Fuchen Zhang; Fei Xu

doi:10.11948/20220364

2024 Volume 14 Issue 4

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Fuchen Zhang, Fei Xu. DYNAMICAL BEHAVIOR OF THE GENERALIZED COMPLEX LORENZ CHAOTIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 1915-1931. doi: 10.11948/20220364

Citation:

Fuchen Zhang, Fei Xu. DYNAMICAL BEHAVIOR OF THE GENERALIZED COMPLEX LORENZ CHAOTIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2024, 14(4): 1915-1931. doi: 10.11948/20220364

DYNAMICAL BEHAVIOR OF THE GENERALIZED COMPLEX LORENZ CHAOTIC SYSTEM

Fuchen Zhang^1, ,,
Fei Xu^2,

1.
Chongqing Key Laboratory of Statistical Intelligent Computing and Monitoring, School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China
2.
Department of Mathematics, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, Canada

Author Bio: Email: fxu.feixu@gmail.com(F. Xu)
Corresponding author: Email: zhangfuchen1983@163.com(F. Zhang)
Fund Project: The authors were supported by the Natural Science Foundation of Chongqing Municipality (Grant No. CSTB2022NSCQ-MSX1548), the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Nos. KJQN202100813, KJQN201800818, KJCX2020037), the National Natural Science Foundation of China (Grant No. 12371137), Scientific and Technological Research Program of Chongqing Technology and Business University (Grant Nos. 1952012, 1952026, 1951075), the Program for Chongqing Key Laboratory of Social Economy and Applied Statistics (Grant No. ZDPTTD201909)

Abstract

Abstract

The purpose of this paper is to investigate the boundedness and global attractivity of the complex Lorenz system:
$\dot x = \alpha \left( {y - x} \right),\dot y = \gamma x - cy - dxz,\dot z = - \beta z + \frac{1}{2}\left( {\bar xy + x\bar y} \right),$
where $ \alpha,\beta,\gamma,c,d $ are real 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 laser physics and rotating fluids dynamics. It is very interesting that we find that this system exhibits chaos phenomenon for the given parameters. Using generalized Lyapunov-like functions, we prove the existence of the ultimate bound set and the globally exponentially attractive set in this generalized complex Lorenz system. The rate of the trajectories is also obtained. Numerical simulations show the effectiveness and correctness of the conclusions. Finally, we present an application of our results that obtained in this paper.
- Complex Lorenz chaotic system /
- chaotic attractor /
- Lyapunov exponent /
- Lyapunov dimension /
- global attractivity
MSC: 34D06, 34H10

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