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The research on the stability theory of fractional order nonlinear system has an important value for the application of synchronization and the control of fractional order chaotic system. The discussion that the stability discrimination of fractional order nonlinear system is converted into that of corresponding integer order nonlinear system has an important significance. In this paper, through the examples, for time-varying coefficient matrix, we point out the existing mistake of the discrimination theorem that states that if the integer system is stable, then its corresponding fractional system with order less than one is also stable. We also analyze the causes of the mistake.
[1] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press)
[2] Chen W C 2008 Chao, Solitions and Fractals 36 1305
[3] Lu J G 2006 Chaos, Solitons and Fractals 27 519
[4] Wang X Y, He Y J 2008 Acta Phys. Sin. 57 1485 (in Chinese) [王兴元, 贺毅杰 2008 物理学报 57 1485]
[5] Si G Q, Sun Z Y, Zhang Y B 2011 Chin. Phys. B 20 080505
[6] Zhang R X, Yang S P 2012 Chin. Phys. B 21 030505
[7] Hu J B, Han Y, Zhao L D 2008 Acta Phys. Sin. 57 7522 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2008 物理学报 57 7522]
[8] Hu J B, Han Y, Zhao L D 2009 Acta Phys. Sin. 58 1441 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2009 物理学报 58 1441]
[9] Hu J B, Han Y, Zhao L D 2009 Acta Phys. Sin. 58 2235 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2009 物理学报 58 2235]
[10] 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]
[11] Deng W H 2007 Journal of Computational Physics 227 1510
[12] Zhu J 1992 System Theory, Proceedings The 24th Southeastern Symposium on and The 3rd Annual Symposium on Communications, Signal Processing Expert Systems, and ASIC VLSI Design Mar1-3, 1992 p355
[13] Matignon D 1996 IMACS SMC Proceedings (Lille: IEEE) p963
[14] Rosenbrock H H 1963 Journal of Electronics and Control 15 73
[15] Liao X X 2010 Theory Methods and Application of Stability (2nd Ed.) (Wuhan: Huazhong University of Science and Technology Press) p57 (in Chinese) [廖晓昕 2010 稳定性的理论、方法和应用(第二版) (武汉:华中科技大学出版社) 第57页]
[16] Deng W H 2010 Nonlinear Analysis 72 1768
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[1] Podlubny I 1999 Fractional Differential Equations (San Diego: Academic Press)
[2] Chen W C 2008 Chao, Solitions and Fractals 36 1305
[3] Lu J G 2006 Chaos, Solitons and Fractals 27 519
[4] Wang X Y, He Y J 2008 Acta Phys. Sin. 57 1485 (in Chinese) [王兴元, 贺毅杰 2008 物理学报 57 1485]
[5] Si G Q, Sun Z Y, Zhang Y B 2011 Chin. Phys. B 20 080505
[6] Zhang R X, Yang S P 2012 Chin. Phys. B 21 030505
[7] Hu J B, Han Y, Zhao L D 2008 Acta Phys. Sin. 57 7522 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2008 物理学报 57 7522]
[8] Hu J B, Han Y, Zhao L D 2009 Acta Phys. Sin. 58 1441 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2009 物理学报 58 1441]
[9] Hu J B, Han Y, Zhao L D 2009 Acta Phys. Sin. 58 2235 (in Chinese) [胡建兵, 韩焱, 赵灵冬 2009 物理学报 58 2235]
[10] 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]
[11] Deng W H 2007 Journal of Computational Physics 227 1510
[12] Zhu J 1992 System Theory, Proceedings The 24th Southeastern Symposium on and The 3rd Annual Symposium on Communications, Signal Processing Expert Systems, and ASIC VLSI Design Mar1-3, 1992 p355
[13] Matignon D 1996 IMACS SMC Proceedings (Lille: IEEE) p963
[14] Rosenbrock H H 1963 Journal of Electronics and Control 15 73
[15] Liao X X 2010 Theory Methods and Application of Stability (2nd Ed.) (Wuhan: Huazhong University of Science and Technology Press) p57 (in Chinese) [廖晓昕 2010 稳定性的理论、方法和应用(第二版) (武汉:华中科技大学出版社) 第57页]
[16] Deng W H 2010 Nonlinear Analysis 72 1768
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