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

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

郑广超 刘崇新 王琰

引用本文:
Citation:

一种具有隐藏吸引子的分数阶混沌系统的动力学分析及有限时间同步

郑广超, 刘崇新, 王琰

Dynamic analysis and finite time synchronization of a fractional-order chaotic system with hidden attractors

Zheng Guang-Chao, Liu Chong-Xin, Wang Yan
PDF
导出引用
  • 对于具有隐藏吸引子的混沌系统,既有文献大多只针对整数阶系统进行分析与控制研究.基于Sprott E系统,构建了仅有一个稳定平衡点的分数阶混沌系统,通过相位图、Poincare映射和功率谱等,分析了该系统的基本动力学特征.结果显示,该系统展现出了丰富而复杂的动力学特性,且通过随阶次变化的分岔图可知,系统在不同阶次下呈现出周期运动、倍周期运动和混沌运动等状态,这些动力学特征对于保密通信等实际工程领域有重要的研究价值.针对该具有隐藏吸引子的分数阶系统,应用分数阶系统有限时间稳定性理论设计控制器,对系统进行有限时间同步控制,并通过数值仿真验证了其有效性.
    Shilnikov criteria believe that the emergence of chaos requires at least one unstable equilibrium, and an attractor is associated with the unstable equilibrium. However, some special chaotic systems have been proposed recently, each of which has one stable equilibrium, or no equilibrium at all, or has a linear equilibrium (infinite equilibrium). These special dynamical systems can present chaotic characteristics, and the attractors in these chaotic systems are called hidden attractors due to the fact that the attraction basins of chaotic systems do not intersect with small neighborhoods of any equilibrium points. Since they were first found and reported in 2011, the dynamical systems with hidden attractors have attracted much attention. Additionally, the fractional-order system, which can give a clearer physical meaning and a more accurate description of the physical phenomenon, has been broadly investigated in recent years. Motivated by these two considerations, in this paper, we propose a fractional-order chaotic system with hidden attractors, and the finite time synchronization of the fractional-order chaotic systems is also studied.Most of the researches mainly focus on dynamic analysis and control of integer-order chaotic systems with hidden attractors. In this paper, based on the Sprott E system, a fractional-order chaotic system is constructed by adding an appropriate constant term. The fractional-order chaotic system has only one stable equilibrium point, but it can generate various hidden attractors. Basic dynamical characteristics of the system are analyzed carefully through phase diagram, Poincare mapping and power spectrum, and the results show that the fractional-order system can present obvious chaotic characteristics. Based on bifurcation diagram of system order, it can be found that the fractional-order system can have period attractors, doubling period attractors, and chaotic attractors with various orders. Additionally, a finite time synchronization of the fractional-order chaotic system with hidden attractors is realized based on the finite time stable theorem, and the proposed controller is robust and can guarantee fast convergence. Finally, numerical simulation is carried out and the results verify the effectiveness of the proposed controller.The fractional-order chaotic system with hidden attractors has more complex and richer dynamic characteristics than integer-order chaotic systems, and chaotic range of parameters is more flexible, meanwhile the dynamics is more sensitive to system parameters. Therefore, the fractional-order chaotic system with hidden attractors can provide more key parameters and present better performance for practical applications, such as secure communication and image encryption, and it deserves to be further investigated.
      通信作者: 郑广超, 342267105@qq.com
    • 基金项目: 国家自然科学基金创新研究群体科学基金(批准号:51521065)资助的课题.
      Corresponding author: Zheng Guang-Chao, 342267105@qq.com
    • Funds: Project supported by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant No. 51521065).
    [1]

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

    [2]

    Rössler O E 1976 Phys. Lett. A 57 397

    [3]

    Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465

    [4]

    Liu C X, Liu T, Liu L, Liu K 2004 Chaos Solitions Fractals 22 1031

    [5]

    L J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659

    [6]

    Liu W B, Chen G R 2003 Int. J. Bifurcation Chaos 13 261

    [7]

    Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295

    [8]

    Bao B C, Liu Z, Xu J P 2009 J. Sys. Eng. Electron. 20 1179

    [9]

    Shilnikov L P 1965 Sov. Math. Dokl. 6 163

    [10]

    Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230

    [11]

    Molaie M, Jafari S, Sprott J C, Golpayegani S M R H 2013 Int. J. Bifurcation Chaos 23 1350188

    [12]

    Wang X, Chen G R 2012 Commu. Nonlinear Sci. Numer. Simul. 17 1264

    [13]

    Jafari S, Sprott J C, Golpayegani S M R H 2013 Phys. Lett. A 377 699

    [14]

    Wei Z 2011 Phys. Lett. A 376 102

    [15]

    Jafari S, Sprott J C 2013 Chaos Solitions Fractals 57 79

    [16]

    Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255

    [17]

    Leonov G A, Kuznetsov N V, Mokaev T N 2015 Commu. Nonlinear Sci. Numer. Simul. 28 166

    [18]

    Zhang Y A, Yu M Z, Wu H L 2016 Acta Electron. Sin. 44 607 (in Chinese) [张友安, 余名哲, 吴华丽 2016 电子学报 44 607]

    [19]

    Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821

    [20]

    Li C G, Liao X F, Yu J B 2003 Phys. Rev. E 68 067203

    [21]

    Jia H Y, Chen Z Q, Yuan Z Z 2010 Chin. Phys. B 19 020507

    [22]

    Zhang L, Yan Y 2014 Nonlinear Dyn. 76 1761

    [23]

    Wang D F, Zhang J Y, Wang X Y 2013 Chin. Phys. B 22 100504

    [24]

    Zhao L D, Hu J B, Bao Z H, Zhang G A, Xu C, Zhang S B 2011 Acta Phys. Sin. 60 100507 (in Chinese) [赵灵冬, 胡建兵, 包志华, 章国安, 徐晨, 张士兵 2011 物理学报 60 100507]

    [25]

    Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) p18

    [26]

    Sprott J C 1994 Phys. Rev. E 50 647

  • [1]

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

    [2]

    Rössler O E 1976 Phys. Lett. A 57 397

    [3]

    Chen G R, Ueta T 1999 Int. J. Bifurcation Chaos 9 1465

    [4]

    Liu C X, Liu T, Liu L, Liu K 2004 Chaos Solitions Fractals 22 1031

    [5]

    L J H, Chen G R 2002 Int. J. Bifurcation Chaos 12 659

    [6]

    Liu W B, Chen G R 2003 Int. J. Bifurcation Chaos 13 261

    [7]

    Qi G Y, Chen G R, Du S Z, Chen Z Q, Yuan Z Z 2005 Physica A 352 295

    [8]

    Bao B C, Liu Z, Xu J P 2009 J. Sys. Eng. Electron. 20 1179

    [9]

    Shilnikov L P 1965 Sov. Math. Dokl. 6 163

    [10]

    Leonov G A, Kuznetsov N V, Vagaitsev V I 2011 Phys. Lett. A 375 2230

    [11]

    Molaie M, Jafari S, Sprott J C, Golpayegani S M R H 2013 Int. J. Bifurcation Chaos 23 1350188

    [12]

    Wang X, Chen G R 2012 Commu. Nonlinear Sci. Numer. Simul. 17 1264

    [13]

    Jafari S, Sprott J C, Golpayegani S M R H 2013 Phys. Lett. A 377 699

    [14]

    Wei Z 2011 Phys. Lett. A 376 102

    [15]

    Jafari S, Sprott J C 2013 Chaos Solitions Fractals 57 79

    [16]

    Li Q D, Zeng H Z, Yang X S 2014 Nonlinear Dyn. 77 255

    [17]

    Leonov G A, Kuznetsov N V, Mokaev T N 2015 Commu. Nonlinear Sci. Numer. Simul. 28 166

    [18]

    Zhang Y A, Yu M Z, Wu H L 2016 Acta Electron. Sin. 44 607 (in Chinese) [张友安, 余名哲, 吴华丽 2016 电子学报 44 607]

    [19]

    Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821

    [20]

    Li C G, Liao X F, Yu J B 2003 Phys. Rev. E 68 067203

    [21]

    Jia H Y, Chen Z Q, Yuan Z Z 2010 Chin. Phys. B 19 020507

    [22]

    Zhang L, Yan Y 2014 Nonlinear Dyn. 76 1761

    [23]

    Wang D F, Zhang J Y, Wang X Y 2013 Chin. Phys. B 22 100504

    [24]

    Zhao L D, Hu J B, Bao Z H, Zhang G A, Xu C, Zhang S B 2011 Acta Phys. Sin. 60 100507 (in Chinese) [赵灵冬, 胡建兵, 包志华, 章国安, 徐晨, 张士兵 2011 物理学报 60 100507]

    [25]

    Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press) p18

    [26]

    Sprott J C 1994 Phys. Rev. E 50 647

  • [1] 高飞, 胡道楠, 童恒庆, 王传美. 分数阶Willis环脑迟发性动脉瘤时滞系统混沌分析. 物理学报, 2018, 67(15): 150501. doi: 10.7498/aps.67.20180262
    [2] 李睿, 张广军, 姚宏, 朱涛, 张志浩. 参数不确定的分数阶混沌系统广义错位延时投影同步. 物理学报, 2014, 63(23): 230501. doi: 10.7498/aps.63.230501
    [3] 王斌, 吴超, 朱德兰. 一个新的分数阶混沌系统的翼倍增及滑模同步. 物理学报, 2013, 62(23): 230506. doi: 10.7498/aps.62.230506
    [4] 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
    [5] 黄丽莲, 何少杰. 分数阶状态空间系统的稳定性分析及其在分数阶混沌控制中的应用. 物理学报, 2011, 60(4): 044703. doi: 10.7498/aps.60.044703
    [6] 刘福才, 李俊义, 臧秀凤. 基于自适应主动及滑模控制的分数阶超混沌系统异结构反同步. 物理学报, 2011, 60(3): 030504. doi: 10.7498/aps.60.030504
    [7] 胡建兵, 章国安, 赵灵冬, 曾金全. 间歇同步分数阶统一混沌系统. 物理学报, 2011, 60(6): 060504. doi: 10.7498/aps.60.060504
    [8] 赵灵冬, 胡建兵, 包志华, 章国安, 徐晨, 张士兵. 分数阶系统有限时间稳定性理论及分数阶超混沌Lorenz系统有限时间同步. 物理学报, 2011, 60(10): 100507. doi: 10.7498/aps.60.100507
    [9] 刘勇, 谢勇. 分数阶FitzHugh-Nagumo模型神经元的动力学特性及其同步. 物理学报, 2010, 59(3): 2147-2155. doi: 10.7498/aps.59.2147
    [10] 阎晓妹, 刘丁. 基于最小二乘支持向量机的分数阶混沌系统控制. 物理学报, 2010, 59(5): 3043-3048. doi: 10.7498/aps.59.3043
    [11] 赵灵冬, 胡建兵, 刘旭辉. 参数未知的分数阶超混沌Lorenz系统的自适应追踪控制与同步. 物理学报, 2010, 59(4): 2305-2309. doi: 10.7498/aps.59.2305
    [12] 张若洵, 杨世平. 分数阶共轭Chen混沌系统中的混沌及其电路实验仿真. 物理学报, 2009, 58(5): 2957-2962. doi: 10.7498/aps.58.2957
    [13] 张若洵, 杨洋, 杨世平. 分数阶统一混沌系统的自适应同步. 物理学报, 2009, 58(9): 6039-6044. doi: 10.7498/aps.58.6039
    [14] 胡建兵, 韩焱, 赵灵冬. 自适应同步参数未知的异结构分数阶超混沌系统. 物理学报, 2009, 58(3): 1441-1445. doi: 10.7498/aps.58.1441
    [15] 胡建兵, 韩焱, 赵灵冬. 一种新的分数阶系统稳定理论及在back-stepping方法同步分数阶混沌系统中的应用. 物理学报, 2009, 58(4): 2235-2239. doi: 10.7498/aps.58.2235
    [16] 赵品栋, 张晓丹. 一类分数阶混沌系统的研究. 物理学报, 2008, 57(5): 2791-2798. doi: 10.7498/aps.57.2791
    [17] 张若洵, 杨世平. 一个分数阶新超混沌系统的同步. 物理学报, 2008, 57(11): 6837-6843. doi: 10.7498/aps.57.6837
    [18] 陈向荣, 刘崇新, 李永勋. 基于非线性观测器的一类分数阶混沌系统完全状态投影同步. 物理学报, 2008, 57(3): 1453-1457. doi: 10.7498/aps.57.1453
    [19] 邵仕泉, 高 心, 刘兴文. 两个耦合的分数阶Chen系统的混沌投影同步控制. 物理学报, 2007, 56(12): 6815-6819. doi: 10.7498/aps.56.6815
    [20] 王发强, 刘崇新. 分数阶临界混沌系统及电路实验的研究. 物理学报, 2006, 55(8): 3922-3927. doi: 10.7498/aps.55.3922
计量
  • 文章访问数:  6339
  • PDF下载量:  523
  • 被引次数: 0
出版历程
  • 收稿日期:  2017-10-31
  • 修回日期:  2017-11-23
  • 刊出日期:  2018-03-05

/

返回文章
返回