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Shimizu-Morioka系统与Finance系统生成Lorenz混沌的微分几何策略

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## Lorenz chaotic system generated from Shimizu-Morioka system or Finance system: Differential geometric approach

Zhang Duan, Shi Jia-Qin, Sun Ying, Yang Xu-Hua, Ye Lei
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• #### 摘要

从一种受控混沌系统生成另一混沌系统可增强保密通信的安全性, 具备潜在应用前景. 研究了如何通过状态变换以及单输入反馈, 驱使受控Shimizu-Morioka系统与受控Finance系统生成Lorenz混沌动态. 主要方法是运用微分几何理论, 将上述三种系统等价转换为下三角形式, 并尽量简化和一致化其方程形式, 使得上述三种不同的3阶系统的前两个方程形式相同, 然后对受控Shimizu-Morioka系统与受控Finance系统设计单输入反馈控制第三个方程的形式, 以便达到生成Lorenz混沌的目的. 运用该方法, 设计了受控Shimizu-Morioka系统通过状态变换和单输入状态反馈, 混沌反控制生成Lorenz混沌的控制策略; 也设计了受控Finance系统通过状态变换和单输入状态反馈, 广义同步到Lorenz混沌的控制策略. 最后, 借助数值仿真验证了上述混沌反控制和广义同步的有效性.

#### Abstract

The problem of how to generate the Lorenz attractor from several nonlinear control systems is investigated in this paper. To be more precise, the conversions from the controlled Shimizu-Morioka system and the controlled Finance system to the Lorenz system are achieved by using the differential geometric control theory. For each case a scalar control input and a state transformation are proposed. The main approach of this paper is to convert all of those three-order systems into so called lower triangular forms which all have the same first two equations. Thus converting the controlled Shimizu-Morioka system or the controlled Finance system into the Lorenz attractor is feasible by choosing an appropriate scalar control input in the third equation of each of the two control systems. To this end, firstly, in order to use the tools of the differential geometry we construct a controlled Lorenz system by treating the vector field of the Lorenz attractor as the drift vector field and treating a linear vector field with three parameters as an input vector field. When those parameters are selected in a special manner, the conditions under which the controlled Lorenz system can be equivalently transformed into the lower triangular form are satisfied. Secondly, a state transformation, through which the controlled Lorenz system can be described as a lower triangular form, is obtained by a method like Gaussian elimination instead of solving three complicated partial differential equations. Employing several partial state transformations, choosing those three parameters and setting a scalar control input, we can reduce the equations of the controlled Lorenz system into its simplest lower triangular form. Thirdly, through two state transformations designed for the controlled Shimizu-Morioka system and the controlled Finance system respectively, the two control systems are converted into their lower triangular forms which are both similar to that of the Lorenz system in a way aforementioned. A smooth scalar controller is given to achieve the anti-control from the controlled Shimizu-Morioka system to the Lorenz attractor while another non-smooth scalar controller is designed to realize the generalized synchronization from the controlled Finance system to the Lorenz system no matter what the initial values of the two systems are. Finally, two numerical simulations demonstrate the control schemes designed in this paper.

#### 作者及机构信息

###### 通信作者: 张端, dzhang@zjut.edu.cn
• 基金项目: 国家自然科学基金(批准号: 61773348)和浙江省自然科学基金(批准号: LY16F030014)资助的课题

#### Authors and contacts

###### Corresponding author: Zhang Duan, dzhang@zjut.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61773348) and the Natural Science Foundation of Zhejiang Province, China (Grant No. LY16F030014)

#### 施引文献

• 图 1  Lorenz系统轨迹

Fig. 1.  Trajectory of the Lorenz system.

图 2  受控Shimizu-Morioka系统轨迹

Fig. 2.  Trajectory of the controlled Shimizu-Morioka system

图 3  受控Shimizu-Morioka系统的标量控制输入

Fig. 3.  Scale control input for the controlled Shimizu-Morioka system.

图 4  经状态变换${{{T}}^{ - 1}}({{\tau }}({{\zeta }}))$受控Shimizu-Morioka系统轨迹

Fig. 4.  Trajectory of the controlled Shimizu-Morioka system via the state transformation ${{{T}}^{ - 1}}({{\tau }}({{\zeta }}))$.

图 5  受控Finance系统轨迹

Fig. 5.  Trajectory of the controlled Finance system.

图 6  受控Finance系统的标量控制输入

Fig. 6.  Scale control input for the controlled Finance system

图 7  经状态变换${{{T}}^{ - 1}}({{\tau }}({{\zeta }}))$受控Finance系统轨迹

Fig. 7.  Trajectory of the controlled Shimizu-Morioka system via the state transformation ${{{T}}^{ - 1}}({{\tau }}({{\zeta }}))$.

图 8  Lorenz系统轨迹与经状态变换${{{T}}^{ - 1}}({{\tau }}({{\zeta }}))$的受控Finance系统的误差

Fig. 8.  Error between the trajectory of the Lorenz system and that of the controlled Finance system via the state transformation ${{{T}}^{ - 1}}({{\tau }}({{\zeta }}))$.