一种异结构分数阶混沌系统投影同步的新方法

黄丽莲; 马楠

doi:10.7498/aps.61.160510

摘要

基于Lyapunov稳定性理论和分数阶系统稳定理论以及分数阶非线性系统性质,提出了一种用来判定分数阶混沌系统是否稳定的新的判定定理,并把该理论运用于对分数阶混沌系统的控制与同步,同时给出了数学证明过程,严格保证了该方法的正确性与一般适用性. 运用所提出的稳定性定理,实现了异结构分数阶混沌系统的投影同步. 对分数阶Lorenz混沌系统与分数阶Liu混沌系统实现了投影同步; 针对四维超混沌分数阶系统,也实现了异结构投影同步. 该稳定性定理避免了求解分数阶平衡点以及Lyapunov指数的问题,从而可以方便地选择出控制律,并且所得的控制器结构简单、适用范围广. 数值仿真的结果取得了预期的效果,进一步验证了这一稳定性定理的正确性及普遍适用性.

关键词:

Abstract

Based on the Lyapunov theory as the breakthrough point, and based on the fractional order system stability theory and properties of fractional nonlinear system, a kind of fractional-order chaotic system is proposed to determine whether the new theorem is stable, and the theory is used for fractional order control and synchronization of chaotic systems, and gives a mathematical proof process to strictly ensure the correctness of the method and general applicability. Then the proposed stability theorem is used to achieve the projective synchronization of fractional Lorenz chaotic system with fractional order chaotic Liu system, as well as the projective synchronization of four-dimensional hyperchaos of fractional order systems of different structures. In the stability theorem solving the fractional balance point and the Lyapunov index are avoided, therefore a control law can be easily selected, and the obtained controller has the advantages of simple structure and wide range of application. Finally, the expected numerical simulation results are achieved, which further proves the correctness and universal applicability of the stability theorem.

Keywords:

作者及机构信息

黄丽莲,
马楠

1.
哈尔滨工程大学信息与通信工程学院, 哈尔滨 150001

基金项目: 国家自然科学基金(批准号: 61172038)和中央高校基本科研业务费 (批准号: HEUCFT1203)资助的课题.

Authors and contacts

1.
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61172038) and the Fundamental Research Fund for the Central Universities, China (Grant No. HEUCFT1203).

参考文献

[1]	Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1193
[2]	Liang C X, Tang J S 2008 Chin. Phys. B 17 135
[3]	Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969
[4]	Li X J, Liu J, Dong P Z, Xing L F 2009 J. Wuhan Univ. Sci. Eng. 22 30
[5]	Qiao Z M, Jin Y R 2010 J. Anhui Univ. (Natural Science Edition) 34 23
[6]	Zhou P, Kuang F 2010 Acta Phys. Sin. 59 6851 (in Chinese) [周平, 邝菲 2010 59 6851]
[7]	Podlubny I 1999 Fractional Differential Equations (New York: Academic Press)
[8]	Mainieri R, Rehacek J 1999 Phys. Rev. Lett. 82 3042
[9]	Li Z G, Xu D 2004 Chaos Soliton. Fract. 22 477
[10]	Chee C Y, Xu D L 2005 Chaos Soliton. Fract. 23 1063
[11]	Wang B H, Bu S L 2004 Int. J. Modern Phys. B 18 2415
[12]	Xu D L 2001 Phys. Rev. E 63 027201
[13]	Wen G L, Xu D L 2005 Chaos Soliton. Fract. 26 71
[14]	Hu M F, Xu Z Y, Zhang R 2008 Commun. Nonlinear Sci. Numer. Simul. 13 456
[15]	Matignon D 1996 IMACS, IEEE-SMC Lille, France, July, 1996 p963
[16]	Hu J B, Han Y, Zhao L D 2009 Acta Phys. Sin. 58 4402 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2009 58 4402]
[17]	Li N, Li J F 2011 Acta Phys. Sin. 60 110512 (in Chinese) [李农, 李建芬 2011 60 110512]
[18]	Lorenz E N 1963 Atmos. Sci. 20 130
[19]	Li C P, Peng G J 2004 Chaos Soliton. Fract. 22 443
[20]	Zhang R X, Yang S P, Liu Y L 2010 Acta Phys. Sin. 59 1549 (in Chinese) [张若洵, 杨世平, 刘永利 2010 59 1549]
[21]	Abdurahman K, Wang X Y, Zhao Y Z 2011 Acta Phys. Sin. 60 040506 (in Chinese) [阿布都热合曼·卡的尔, 王兴元, 赵玉章 2011 60 040506]
[22]	Liu C X, Liu T, Liu L, Liu K 2004 Chaos Soliton. Fract. 22 1031

施引文献

[1]	Ott E, Grebogi C, Yorke J A 1990 Phys. Rev. Lett. 64 1193
[2]	Liang C X, Tang J S 2008 Chin. Phys. B 17 135
[3]	Zhang H G, Fu J, Ma T D, Tong S C 2009 Chin. Phys. B 18 969
[4]	Li X J, Liu J, Dong P Z, Xing L F 2009 J. Wuhan Univ. Sci. Eng. 22 30
[5]	Qiao Z M, Jin Y R 2010 J. Anhui Univ. (Natural Science Edition) 34 23
[6]	Zhou P, Kuang F 2010 Acta Phys. Sin. 59 6851 (in Chinese) [周平, 邝菲 2010 59 6851]
[7]	Podlubny I 1999 Fractional Differential Equations (New York: Academic Press)
[8]	Mainieri R, Rehacek J 1999 Phys. Rev. Lett. 82 3042
[9]	Li Z G, Xu D 2004 Chaos Soliton. Fract. 22 477
[10]	Chee C Y, Xu D L 2005 Chaos Soliton. Fract. 23 1063
[11]	Wang B H, Bu S L 2004 Int. J. Modern Phys. B 18 2415
[12]	Xu D L 2001 Phys. Rev. E 63 027201
[13]	Wen G L, Xu D L 2005 Chaos Soliton. Fract. 26 71
[14]	Hu M F, Xu Z Y, Zhang R 2008 Commun. Nonlinear Sci. Numer. Simul. 13 456
[15]	Matignon D 1996 IMACS, IEEE-SMC Lille, France, July, 1996 p963
[16]	Hu J B, Han Y, Zhao L D 2009 Acta Phys. Sin. 58 4402 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2009 58 4402]
[17]	Li N, Li J F 2011 Acta Phys. Sin. 60 110512 (in Chinese) [李农, 李建芬 2011 60 110512]
[18]	Lorenz E N 1963 Atmos. Sci. 20 130
[19]	Li C P, Peng G J 2004 Chaos Soliton. Fract. 22 443
[20]	Zhang R X, Yang S P, Liu Y L 2010 Acta Phys. Sin. 59 1549 (in Chinese) [张若洵, 杨世平, 刘永利 2010 59 1549]
[21]	Abdurahman K, Wang X Y, Zhao Y Z 2011 Acta Phys. Sin. 60 040506 (in Chinese) [阿布都热合曼·卡的尔, 王兴元, 赵玉章 2011 60 040506]
[22]	Liu C X, Liu T, Liu L, Liu K 2004 Chaos Soliton. Fract. 22 1031

[1]	陈晔, 李生刚, 刘恒. 基于自适应模糊控制的分数阶混沌系统同步. , 2016, 65(17): 170501. doi: 10.7498/aps.65.170501
[2]	杨叶红, 肖剑, 马珍珍. 部分线性的分数阶混沌系统修正函数投影同步. , 2013, 62(18): 180505. doi: 10.7498/aps.62.180505
[3]	王春华, 胡燕, 余飞, 徐浩. 一类混沌系统同步时间可控的自适应投影同步. , 2013, 62(11): 110509. doi: 10.7498/aps.62.110509
[4]	黄丽莲, 齐雪. 基于自适应滑模控制的不同维分数阶混沌系统的同步. , 2013, 62(8): 080507. doi: 10.7498/aps.62.080507
[5]	马铁东, 江伟波, 浮洁, 柴毅, 陈立平, 薛方正. 一类分数阶混沌系统的自适应同步. , 2012, 61(16): 160506. doi: 10.7498/aps.61.160506
[6]	马铁东, 江伟波, 浮洁. 基于比较系统方法的分数阶混沌系统脉冲同步控制. , 2012, 61(9): 090503. doi: 10.7498/aps.61.090503
[7]	吕翎, 孟乐, 郭丽, 邹家蕊, 杨明. 激光时空混沌模型的加权网络投影同步. , 2011, 60(3): 030506. doi: 10.7498/aps.60.030506
[8]	曹鹤飞, 张若洵. 基于滑模控制的分数阶混沌系统的自适应同步. , 2011, 60(5): 050510. doi: 10.7498/aps.60.050510
[9]	孙宁, 张化光, 王智良. 基于分数阶滑模面控制的分数阶超混沌系统的投影同步. , 2011, 60(5): 050511. doi: 10.7498/aps.60.050511
[10]	胡建兵, 肖建, 赵灵冬. 阶次不等的分数阶混沌系统同步. , 2011, 60(11): 110515. doi: 10.7498/aps.60.110515
[11]	孙宁. 基于区间系统理论的分数阶混沌系统同步. , 2011, 60(12): 120506. doi: 10.7498/aps.60.120506
[12]	许喆, 刘崇新, 杨韬. 基于Lyapunov方程的分数阶新混沌系统的控制. , 2010, 59(3): 1524-1531. doi: 10.7498/aps.59.1524
[13]	周平, 邝菲. 分数阶混沌系统与整数阶混沌系统之间的同步. , 2010, 59(10): 6851-6858. doi: 10.7498/aps.59.6851
[14]	孟娟, 王兴元. 基于模糊观测器的Chua混沌系统投影同步. , 2009, 58(2): 819-823. doi: 10.7498/aps.58.819
[15]	刘丁, 闫晓妹. 基于滑模控制实现分数阶混沌系统的投影同步. , 2009, 58(6): 3747-3752. doi: 10.7498/aps.58.3747
[16]	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. , 2008, 57(3): 1453-1457. doi: 10.7498/aps.57.1453
[17]	王兴元, 贺毅杰. 分数阶统一混沌系统的投影同步. , 2008, 57(3): 1485-1492. doi: 10.7498/aps.57.1485
[18]	胡建兵, 韩焱, 赵灵冬. 基于Lyapunov方程的分数阶混沌系统同步. , 2008, 57(12): 7522-7526. doi: 10.7498/aps.57.7522
[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. , 2007, 56(12): 6815-6819. doi: 10.7498/aps.56.6815
[20]	刘杰, 陈士华, 陆君安. 统一混沌系统的投影同步与控制. , 2003, 52(7): 1595-1599. doi: 10.7498/aps.52.1595

计量

文章访问数: 7015
PDF下载量: 595
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板