| Citation: |
Fuchen Zhang, Song Chen, Xiusu Chen, Jinde Cao, Fei Xu. SYNCHRONIZATION OF A NOVEL COMPLEX SYSTEM[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2508-2527. doi: 10.11948/20240020
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SYNCHRONIZATION OF A NOVEL COMPLEX SYSTEM
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Abstract
In this paper, we introduce a new hyperchaotic system which is a four-dimensional system of nonlinear differential equations. This system exhibits very rich chaotic dynamical behaviors. The dynamical characteristics of this system are analysed theoretically and numerically, including the dissipation, the quilibrium points and their stability, Lyapunov exponents, the Lyapunov dimension of the attractors, the global exponential attractive set. In order to achieve synchronization fast, global exponential synchronization is adopted. Suitable linear and nonlinear controllers have been designed to achieve global exponential synchronization between two identical chaotic systems by using the Lyapunov stability theory and Dini derivative. The innovation of this paper is that firstly we get the globally exponential attractive set of this system. Secondly, the result of globally exponential attractive set of the chaotic system is applied to chaos synchronization. Thirdly, we can get the precise lower bound of the coefficient of linear feedback controller ${k_1}, {k_2}, {k_3}$ and ${k_4}$. Finally, chaos synchronization is studied numerically. Numerical simulations are in excellent agreement with the theoretical study.
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References
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Fuchen Zhang, Song Chen, Xiusu Chen, Jinde Cao, Fei Xu. SYNCHRONIZATION OF A NOVEL COMPLEX SYSTEM[J]. Journal of Applied Analysis & Computation, 2025, 15(5): 2508-2527. doi: 10.11948/20240020
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Figure 1.
The hyperchaotic attractors of system (2.2) in the 3D space
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Figure 2.
The Lyapunov exponent chart of system (2.2)
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Figure 3.
Lyapunov exponents diagram of system with
$ a \in \left[ {0, 100} \right] $ -
Figure 4.
Bifurcation diagram of state variable
$ x $ $ a $ -
Figure 5.
Lyapunov exponents diagram of system (2.2) with
$ b \in [0, 100]. $ -
Figure 6.
Bifurcation diagram of state variable
$ x $ $ b $ -
Figure 7.
Synchronization error of linear feedback control is illustrated when
$ {k_1} = 38, {k_2} = 72, {k_3} = 5160, {k_4} = 5. $ -
Figure 8.
Synchronization error of nonlinear feedback control is illustrated when
$ {l_1} = - 33, {l_2} = 1, {l_3} = - 1, {l_4} = 1. $