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In this paper, a sliding mode controller with fractional sliding surface is designed for projective synchronization (PS) of fractional hyperchaotic systems via active control theory and sliding mode theory. The existence and the stability of PS between two fractional hyperchaotic systems are studied based on Lyapunov theory, fractional stability theory and the theory of fractional calculus. The criterions for the practical stability of the PS error system are presented also. Numerical simulation of PS between fractional Chen system and a new hyperchaotic system shows the effectiveness of the proposed controller law.
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Keywords:
- sliding mode controller with fractional sliding surface /
- stability analysis /
- fractional hyperchaotic system /
- projective synchronization
[1] Zhang H G, Zhao Y, Yu W, Yang D S 2008 Chin. Phys. B 17 4056
[2] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[3] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3742
[4] Mohammad S T, Mohammad H 2008 Physica A 387 57
[5] Yang D S, Zhang H G, Li A P, Meng Z Y 2007 Acta Phys. Sin. 56 3121 (in Chinese)[杨东升、张化光、李爱平、孟子怡 2007 56 3121]
[6] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3751
[7] Meng J,Wang X Y 2007 Acta Phys. Sin.56 5124 (in Chinese) [孟 娟、王兴元 2007 56 5142]
[8] Zhang H G, Liu D R, Wang Z L 2009 Controlling Chaos: Suppression, Synchronization and Chaotification ( New York :Springer Verlag) p184
[9] Ma T D, Zhang H G, Wang Z L 2007 Acta Phys. Sin. 56 3796 (in Chinese) [马铁东、张化光、王智良 2007 56 3796]
[10] Wang X Y, Wu X J 2007 Acta Phys. Sin. 56 1988 (in Chinese)[王兴元、武相军 2007 56 1988]
[11] Wang X Y, He Y J 2008 Acta Phys. Sin. 57 1485 (in Chinese)[王兴元、贺毅杰 2008 57 1485]
[12] Liu D, Yan X M 2009 Acta Phys. Sin. 58 3747 (in Chinese)[刘 丁、闫晓妹 2009 58 3747]
[13] Mainieri R, Rehacek J 1999 Phys. Rev. Lett. 82 3042
[14] Wu X J, Lu H T, Shen S L 2009 Phys. Lett. A 373 2329
[15] Li C G, Chen G R 2004 Chaos, Solitons & Fractals 22 549
[16] Yu Y G, Li H X 2008 Physica A 387 1393
[17] Liu J, Li X J IEEE, 2008 7th World Congress on Intelligent Control and Automation Chongqing, People R China) 2008 p4653
[18] Si-Ammour A, Djennoune S, Bettayeb M 2009 Commun Nonlinear Sci Numer Simulat 14 2310
[19] Hartley T T, Lorenzo C F, Qammer H K 1995 IEEE Trans. CAS-I 42 485
[20] Podlubny I 1999 Fractional differential equations (San Diego : Academic Press) p18
[21] Matignon D 1996 IMACS, IEEE-SMC Proceeding Lille, France, p963
[22] Khalil H K 1996 Nonlinear systems (Upper Saddle River, NJ:Prentice-Hall) p63
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[1] Zhang H G, Zhao Y, Yu W, Yang D S 2008 Chin. Phys. B 17 4056
[2] Pecora L M, Carroll T L 1990 Phys. Rev. Lett. 64 821
[3] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3742
[4] Mohammad S T, Mohammad H 2008 Physica A 387 57
[5] Yang D S, Zhang H G, Li A P, Meng Z Y 2007 Acta Phys. Sin. 56 3121 (in Chinese)[杨东升、张化光、李爱平、孟子怡 2007 56 3121]
[6] Zhang H G, Ma T D, Fu J, Tong S C 2009 Chin. Phys. B 18 3751
[7] Meng J,Wang X Y 2007 Acta Phys. Sin.56 5124 (in Chinese) [孟 娟、王兴元 2007 56 5142]
[8] Zhang H G, Liu D R, Wang Z L 2009 Controlling Chaos: Suppression, Synchronization and Chaotification ( New York :Springer Verlag) p184
[9] Ma T D, Zhang H G, Wang Z L 2007 Acta Phys. Sin. 56 3796 (in Chinese) [马铁东、张化光、王智良 2007 56 3796]
[10] Wang X Y, Wu X J 2007 Acta Phys. Sin. 56 1988 (in Chinese)[王兴元、武相军 2007 56 1988]
[11] Wang X Y, He Y J 2008 Acta Phys. Sin. 57 1485 (in Chinese)[王兴元、贺毅杰 2008 57 1485]
[12] Liu D, Yan X M 2009 Acta Phys. Sin. 58 3747 (in Chinese)[刘 丁、闫晓妹 2009 58 3747]
[13] Mainieri R, Rehacek J 1999 Phys. Rev. Lett. 82 3042
[14] Wu X J, Lu H T, Shen S L 2009 Phys. Lett. A 373 2329
[15] Li C G, Chen G R 2004 Chaos, Solitons & Fractals 22 549
[16] Yu Y G, Li H X 2008 Physica A 387 1393
[17] Liu J, Li X J IEEE, 2008 7th World Congress on Intelligent Control and Automation Chongqing, People R China) 2008 p4653
[18] Si-Ammour A, Djennoune S, Bettayeb M 2009 Commun Nonlinear Sci Numer Simulat 14 2310
[19] Hartley T T, Lorenzo C F, Qammer H K 1995 IEEE Trans. CAS-I 42 485
[20] Podlubny I 1999 Fractional differential equations (San Diego : Academic Press) p18
[21] Matignon D 1996 IMACS, IEEE-SMC Proceeding Lille, France, p963
[22] Khalil H K 1996 Nonlinear systems (Upper Saddle River, NJ:Prentice-Hall) p63
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