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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.
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Keywords:
- fractional-order chaotic systems /
- stability /
- Lyapunov theory /
- projective synchronization
[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 物理学报 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 物理学报 58 4402]
[17] Li N, Li J F 2011 Acta Phys. Sin. 60 110512 (in Chinese) [李农, 李建芬 2011 物理学报 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 物理学报 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 物理学报 60 040506]
[22] Liu C X, Liu T, Liu L, Liu K 2004 Chaos Soliton. Fract. 22 1031
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[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 物理学报 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 物理学报 58 4402]
[17] Li N, Li J F 2011 Acta Phys. Sin. 60 110512 (in Chinese) [李农, 李建芬 2011 物理学报 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 物理学报 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 物理学报 60 040506]
[22] Liu C X, Liu T, Liu L, Liu K 2004 Chaos Soliton. Fract. 22 1031
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