搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

张端 施佳琴 孙莹 杨旭华 叶蕾

引用本文:
Citation:

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

张端, 施佳琴, 孙莹, 杨旭华, 叶蕾

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
PDF
HTML
导出引用
  • 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.
    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.
      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
      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)
    [1]

    Boccaletti S, Grebogi C, Lai Y, Mancini H L, Maza D 2000 Phys. Rep. 329 103Google Scholar

    [2]

    Alves P R, Duarte L G, Mota L A 2018 Chaos, Solitons Fractals 110 232Google Scholar

    [3]

    Moskalenko O I, Koronovskii A A, Hramov A E 2010 Phys. Lett. A 374 2925Google Scholar

    [4]

    Pang M, Huang Y 2018 Chin. Phys. B 27 118902Google Scholar

    [5]

    Wang J, Duan L, Kang Q 2017 Chin. Phys. Lett. 34 074703Google Scholar

    [6]

    Moreno-Valenzuela J 2013 Commun. Nonlinear Sci. 18 1Google Scholar

    [7]

    Yan S 2016 Chin. Phys. B 25 90504Google Scholar

    [8]

    王芳, 张新政, 申朝文, 禹思敏 2012 物理学报 61 190505Google Scholar

    Wang F, Zhang X Z, Shen C W, Yu S M 2012 Acta Phys. Sin. 61 190505Google Scholar

    [9]

    Pecora L M, Carroll T L 2015 Chaos 25 097611Google Scholar

    [10]

    Rulkov N F, Sushchik M M, Tsimring L S, Abarbanel H D 1995 Phys. Rev. E 51 980Google Scholar

    [11]

    Abarbanel H D I, Rulkov N F, Sushchik M M 1996 Phys. Rev. E 53 4528

    [12]

    Parlitz U, Junge L, Kocarev L 1997 Phys. Rev. Lett. 79 3158Google Scholar

    [13]

    Zang H Y, Min L Q, Zhao G, Chen G R 2013 Chin. Phys. Lett. 30 040502Google Scholar

    [14]

    Wu X, Guan Z, Wu Z, Li T 2007 Phys. Lett. A 364 484Google Scholar

    [15]

    Ojo K S, Ogunjo S T, Njah A N, Fuwape I 2015 Pramana-J. Phys. 84 33Google Scholar

    [16]

    Zhang G, Liu Z, Ma Z 2007 Chaos, Solitons Fractals 32 773Google Scholar

    [17]

    Wang X F, Chen G 2003 IEEE Trans. Circ. Syst. I 50 812Google Scholar

    [18]

    Zhang J, Li C, Zhang H, Yu J 2004 Chaos, Solitons Fractals 21 1183Google Scholar

    [19]

    Wang F, Chen S, Yu M, Wang C 2004 Chaos, Solitons Fractals 22 927Google Scholar

    [20]

    Hramov A E, Koronovskii A A 2005 Phys. Rev. E 71 067201Google Scholar

    [21]

    Hu A, Xu Z, Guo L 2009 Phys. Lett. A 373 2319Google Scholar

    [22]

    Zhou J, Chen J, Lu J, Lü J 2017 IEEE Trans. Automat. Contr. 62 3468Google Scholar

    [23]

    Shimizu T, Morioka N 1980 Phys. Lett. A 76 201Google Scholar

    [24]

    Ma C, Wang X 2012 Commun. Nonlinear Sci. 17 721Google Scholar

    [25]

    Lorenz E N 1963 J. Atmos. Sci. 20 130Google Scholar

    [26]

    Celikovsky S, Nijmeijer H 1996 Syst. Control Lett. 27 135Google Scholar

    [27]

    Khalil H K 2002 Nonlinear Systems (Londan: Prentice Hall) pp521−530

    [28]

    Fecko M 2006 Differential Geometry and Lie Groups for Physicists (Cambridge: Cambridge University Press) pp21−91, 524−550

    [29]

    Haimo V T 1986 SIAM J. Control Optim. 24 760Google Scholar

  • 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 ${{{T}}^{ - 1}}({{\tau }}({{\zeta }}))$.

  • [1]

    Boccaletti S, Grebogi C, Lai Y, Mancini H L, Maza D 2000 Phys. Rep. 329 103Google Scholar

    [2]

    Alves P R, Duarte L G, Mota L A 2018 Chaos, Solitons Fractals 110 232Google Scholar

    [3]

    Moskalenko O I, Koronovskii A A, Hramov A E 2010 Phys. Lett. A 374 2925Google Scholar

    [4]

    Pang M, Huang Y 2018 Chin. Phys. B 27 118902Google Scholar

    [5]

    Wang J, Duan L, Kang Q 2017 Chin. Phys. Lett. 34 074703Google Scholar

    [6]

    Moreno-Valenzuela J 2013 Commun. Nonlinear Sci. 18 1Google Scholar

    [7]

    Yan S 2016 Chin. Phys. B 25 90504Google Scholar

    [8]

    王芳, 张新政, 申朝文, 禹思敏 2012 物理学报 61 190505Google Scholar

    Wang F, Zhang X Z, Shen C W, Yu S M 2012 Acta Phys. Sin. 61 190505Google Scholar

    [9]

    Pecora L M, Carroll T L 2015 Chaos 25 097611Google Scholar

    [10]

    Rulkov N F, Sushchik M M, Tsimring L S, Abarbanel H D 1995 Phys. Rev. E 51 980Google Scholar

    [11]

    Abarbanel H D I, Rulkov N F, Sushchik M M 1996 Phys. Rev. E 53 4528

    [12]

    Parlitz U, Junge L, Kocarev L 1997 Phys. Rev. Lett. 79 3158Google Scholar

    [13]

    Zang H Y, Min L Q, Zhao G, Chen G R 2013 Chin. Phys. Lett. 30 040502Google Scholar

    [14]

    Wu X, Guan Z, Wu Z, Li T 2007 Phys. Lett. A 364 484Google Scholar

    [15]

    Ojo K S, Ogunjo S T, Njah A N, Fuwape I 2015 Pramana-J. Phys. 84 33Google Scholar

    [16]

    Zhang G, Liu Z, Ma Z 2007 Chaos, Solitons Fractals 32 773Google Scholar

    [17]

    Wang X F, Chen G 2003 IEEE Trans. Circ. Syst. I 50 812Google Scholar

    [18]

    Zhang J, Li C, Zhang H, Yu J 2004 Chaos, Solitons Fractals 21 1183Google Scholar

    [19]

    Wang F, Chen S, Yu M, Wang C 2004 Chaos, Solitons Fractals 22 927Google Scholar

    [20]

    Hramov A E, Koronovskii A A 2005 Phys. Rev. E 71 067201Google Scholar

    [21]

    Hu A, Xu Z, Guo L 2009 Phys. Lett. A 373 2319Google Scholar

    [22]

    Zhou J, Chen J, Lu J, Lü J 2017 IEEE Trans. Automat. Contr. 62 3468Google Scholar

    [23]

    Shimizu T, Morioka N 1980 Phys. Lett. A 76 201Google Scholar

    [24]

    Ma C, Wang X 2012 Commun. Nonlinear Sci. 17 721Google Scholar

    [25]

    Lorenz E N 1963 J. Atmos. Sci. 20 130Google Scholar

    [26]

    Celikovsky S, Nijmeijer H 1996 Syst. Control Lett. 27 135Google Scholar

    [27]

    Khalil H K 2002 Nonlinear Systems (Londan: Prentice Hall) pp521−530

    [28]

    Fecko M 2006 Differential Geometry and Lie Groups for Physicists (Cambridge: Cambridge University Press) pp21−91, 524−550

    [29]

    Haimo V T 1986 SIAM J. Control Optim. 24 760Google Scholar

  • [1] 杨红丽, 刘楠, 杨联贵. Mdm2介导的正反馈环对p53基因网络振荡行为的影响. 物理学报, 2021, 70(13): 138701. doi: 10.7498/aps.70.20210015
    [2] 黄沄. 一类多翼蝴蝶混沌吸引子及其电路实现. 物理学报, 2014, 63(8): 080505. doi: 10.7498/aps.63.080505
    [3] 艾星星, 孙克辉, 贺少波, 王会海. 简化Lorenz多涡卷混沌吸引子的设计与应用. 物理学报, 2014, 63(12): 120511. doi: 10.7498/aps.63.120511
    [4] 张立森, 蔡理, 冯朝文. 线性延时反馈Josephson结的Hopf分岔和混沌化. 物理学报, 2011, 60(6): 060306. doi: 10.7498/aps.60.060306
    [5] 尚慧琳. 时滞位移反馈对Helmholtz振子系统的分形侵蚀安全域的控制. 物理学报, 2011, 60(7): 070501. doi: 10.7498/aps.60.070501
    [6] 赵艳影, 杨如铭. 利用时滞反馈控制自参数振动系统饱和控制减振频带. 物理学报, 2011, 60(10): 104304. doi: 10.7498/aps.60.104304.2
    [7] 杨红, 王瑞. 基于反馈和多最小二乘支持向量机的分数阶混沌系统控制. 物理学报, 2011, 60(7): 070508. doi: 10.7498/aps.60.070508
    [8] 谭平安, 张波, 丘东元. 晶闸管混沌行为的延迟反馈控制与尖峰电流抑制. 物理学报, 2010, 59(8): 5299-5306. doi: 10.7498/aps.59.5299
    [9] 史正平. 简易混沌振荡器的混沌特性及其反馈控制电路的设计. 物理学报, 2010, 59(9): 5940-5948. doi: 10.7498/aps.59.5940
    [10] 张莹, 雷佑铭, 方同. 混沌吸引子的对称破缺激变. 物理学报, 2009, 58(6): 3799-3805. doi: 10.7498/aps.58.3799
    [11] 孔令琴, 樊林林, 王安帮, 王云才. 相干长度可连续调谐的半导体激光器. 物理学报, 2009, 58(11): 7680-7685. doi: 10.7498/aps.58.7680
    [12] 程桂平, 郑 俊, 邓文武, 李高翔. 反馈法定位两原子之间的相对位置. 物理学报, 2008, 57(1): 212-218. doi: 10.7498/aps.57.212
    [13] 孔令琴, 王安帮, 王海红, 王云才. 光反馈半导体激光器产生低频起伏与高维混沌信号及其演化过程. 物理学报, 2008, 57(4): 2266-2272. doi: 10.7498/aps.57.2266
    [14] 卢伟国, 周雒维, 罗全明, 杜 雄. BOOST变换器延迟反馈混沌控制及其优化. 物理学报, 2007, 56(11): 6275-6281. doi: 10.7498/aps.56.6275
    [15] 陶朝海, 陆君安. 混沌系统的速度反馈同步. 物理学报, 2005, 54(11): 5058-5061. doi: 10.7498/aps.54.5058
    [16] 唐国宁, 罗晓曙. 混沌系统的预测反馈控制. 物理学报, 2004, 53(1): 15-20. doi: 10.7498/aps.53.15
    [17] 陶朝海, 陆君安. 统一混沌系统的控制. 物理学报, 2003, 52(2): 281-284. doi: 10.7498/aps.52.281
    [18] 邹艳丽, 罗晓曙, 方锦清, 汪秉宏. 脉冲电压微分反馈法控制buck功率变换器中的混沌. 物理学报, 2003, 52(12): 2978-2984. doi: 10.7498/aps.52.2978
    [19] 伍维根, 古天祥. 混沌系统的非线性反馈跟踪控制. 物理学报, 2000, 49(10): 1922-1925. doi: 10.7498/aps.49.1922
    [20] 李国辉, 周世平, 徐得名, 赖建文. 间隙线性反馈控制混沌. 物理学报, 2000, 49(11): 2123-2128. doi: 10.7498/aps.49.2123
计量
  • 文章访问数:  4922
  • PDF下载量:  51
  • 被引次数: 0
出版历程
  • 收稿日期:  2019-06-13
  • 修回日期:  2019-09-15
  • 上网日期:  2019-11-27
  • 刊出日期:  2019-12-01

/

返回文章
返回