# 具有隐藏吸引子的广义Lorenz系统动力学分析Dynamics Analysis of Generalized Lorenz System with Hidden Attractors

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In this paper, a generalized Lorenz system with hidden attractors is introduced. The dynamic phenomena including periodic motion, chaotic state and transient chaos is observed when different initial states are given. It is found that the dynamic behavior of the generalized Lorenz system depends on the system parameters and initial states. Through a series of calculations, the Hamiltonian energy function of the system is given, and the energy conversion in different motion states is analyzed.

1. 引言

2. 具有隐藏吸引子的广义Lorenz系统

$\left\{\begin{array}{l}\stackrel{˙}{x}=-a\left(x-y\right)-byz\\ \stackrel{˙}{y}=cx-y-xz\\ \stackrel{˙}{z}=-dz+xy\end{array}$ (1)

3. 具有隐藏吸引子的广义Lorenz系统的Hamilton能量

$\stackrel{˙}{x}=f\left(x\right)$ (2)

$f\left(x\right)={f}_{c}\left(x\right)+{f}_{d}\left(x\right)$ (3)

$\nabla {H}^{\text{T}}{f}_{c}\left(x\right)=0$ (4)

$\stackrel{˙}{H}=\frac{\text{d}H}{\text{d}t}=\nabla {H}^{\text{T}}{f}_{d}\left(x\right)$ (5)

${f}_{c}\left(x\right)=\left(\begin{array}{c}ay-byz\\ cx-xz\\ xy\end{array}\right)，\text{\hspace{0.17em}}{f}_{d}\left(x\right)=\left(\begin{array}{c}-ax\\ -y\\ -dz\end{array}\right)$ (6)

$\left(ay-byz\right)\frac{\partial H}{\partial x}+\left(cx-xz\right)\frac{\partial H}{\partial y}+xy\frac{\partial H}{\partial z}=0$ (7)

$H=\frac{1}{2}\left[{x}^{2}-\frac{a}{c}{y}^{2}+\left(b-\frac{a}{c}\right){z}^{2}\right]$ (8)

Hamilton能量随时间的变化情况为：

$\begin{array}{c}\stackrel{˙}{H}=\frac{1}{2}\cdot 2x\cdot \stackrel{˙}{x}-\frac{a}{2c}\cdot 2y\cdot \stackrel{˙}{y}+\left(\frac{b}{2}-\frac{a}{2c}\right)\cdot 2z\cdot \stackrel{˙}{z}\\ =x\cdot \stackrel{˙}{x}-\frac{a}{c}\cdot y\cdot \stackrel{˙}{y}+\left(b-\frac{a}{c}\right)\cdot z\cdot \stackrel{˙}{z}\\ =x\left[-a\left(x-y\right)-byz\right]-\frac{a}{c}\cdot y\left(cx-y-xa\right)+\left(b-\frac{a}{c}\right)\cdot z\left(-dz+xy\right)\\ =-a{x}^{2}+axy-bxyz-axy+\frac{a}{c}{y}^{2}+\frac{a}{c}xyz-db{z}^{2}+bxyz+\frac{ad}{c}{z}^{2}-\frac{a}{c}xyz\end{array}$

$\begin{array}{l}=-a{x}^{2}+\frac{a}{c}{y}^{2}+\left(db{z}^{2}+\frac{ad}{c}\right){z}^{2}\\ =x\cdot \left(-ax\right)+\left(-\frac{a}{c}y\right)\cdot \left(-y\right)+\left(b-\frac{a}{c}\right)\cdot z\cdot \left(-dz\right)\\ =\nabla {H}^{\text{T}}{f}_{d}\end{array}$

4. 广义Lorenz系统的数值模拟

(a) 选取参数 $a=10$$b=0$$c=24.5$$d=8/3$

Figure 1. The evolution of chaotic attractors in system(1)

Figure 2. The time response of state variable x(t) and Hamilton energy H

(b) 选取参数 $a=1.5$$b=-0.5$$c=3$$d=1$

Figure 3. The time response of state variable x(t) and Hamilton energy H

(c) 选取参数 $a=2.98$$b=-0.438$$c=6.8$$d=1$

Figure 4. The time response of state variable x(t) and Hamilton energy H

Figure 5. The part enlarged picture of Figure 4 and evolution process of system (1)

5. 结束语

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