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Based on the stability theory of fractional order linear systems, the dynamic behavior of the fractional order Newton-Leipnik system with double attractor is studied. Our research shows that the fractional order Newton-Leipnik system involves reverse Hopf bifurcation course, i.e., with the decrease of fractional order, the fractional order Newton-Leipnik system shows mutation from double attractor to single attractor, the dynamic behavior experiences chaos, transient period and converges to one stable equilibrium.
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Keywords:
- Newton-Leipnik system /
- fractional order /
- double attractors
[1] ［1］Podlubny I 1999 Fractional differential equations （New York: Academic Press）
[2] ［2］Hilfer R 2001 Applications of fractional calculus in physics （New Jersey: World Scientific）
[3] ［3］Bagley R L, Calico R A 1991 J. Guid. Control Dynam. 14 304
[4] ［4］Koeller R C 1984 J. Appl. Mech. 51 294
[5] ［5］Koeller R C 1986 Acta Mechanica 58 251
[6] ［6］Heaviside O 1971 Electromagnetic theory （New York: Chelsea）
[7] ［7］Grigorenko I, Grigorenko E 2003 Phys. Rev. Lett. 91 034101
[8] ［8］Gao X, Yu J B 2005 Chaos, Solitons Fract. 24 1097
[9] ［9］Li C G, Chen G 2004 Chaos, Solitons Fract. 22 549
[10] ］Li C P, Peng G J 2004 Chaos, Solitons Fract. 22 443
[11] ］Gao X, Yu J B 2005 Chin. Phys. 14 908
[12] ］Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922 （in Chinese）［王发强，刘崇新 2006 55 3922］
[13] ］Liu C X 2007 Acta Phys. Sin. 56 6865 （in Chinese）［刘崇新 2007 56 6865］
[14] ］Chen X R, Liu C X, Wang F Q, Li Y X 2008 Acta Phys. Sin. 57 1416 （in Chinese）［陈向荣、刘崇新、王发强、李永勋2008 57 1416］
[15] ］Zhang R X, Yang S P 2009 Acta Phys. Sin. 58 2957 （in Chinese）［张若洵、杨世平2009 58 2957］
[16] ］Leipnik R B, Newton T A 1981 Phys. Lett. A 86 63
[17] ］Caputo M 1967 Goephys. J. R. Atr. Soc. 13 529
[18] ］Matignon D 1996 Computational Eng. in Sys. Appl. 12 963
[19] ］Feigenbaum M J 1978 J. Stat. Phys. 19 25
[20] ］Alexander J 1988 Dynamical systems （New York: Springer）
[21] ］Kan I, Kocak H, Yorke J 1992 Annals Math. 136 219
[22] ］Dafilis M P, Bourke P D, Liley D T J, Cadusch P J 2002 Comput. Graph. 26 971
[23] ］Wang X Y, Luo C 2006 Appl. Math. Comput. 183 30
[24] ］Wang X Y 2003 Chaos in the Complex Nonlinearity System （Bejjing: Electronics Industry Press） （in Chinese）［王兴元2003 复杂非线性系统中的混沌（北京：电子工业出版社）
[25] ］Kuruvilla T, Nandakumaran V M 1999 Phys. Lett. A 254 59
[26] ］Sheu L J, Chen H K, Chen J H, Tam L M, Chen W C, Lin K T, Kang Y 2008 Chaos, Solitons Fract. 36 98
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[1] ［1］Podlubny I 1999 Fractional differential equations （New York: Academic Press）
[2] ［2］Hilfer R 2001 Applications of fractional calculus in physics （New Jersey: World Scientific）
[3] ［3］Bagley R L, Calico R A 1991 J. Guid. Control Dynam. 14 304
[4] ［4］Koeller R C 1984 J. Appl. Mech. 51 294
[5] ［5］Koeller R C 1986 Acta Mechanica 58 251
[6] ［6］Heaviside O 1971 Electromagnetic theory （New York: Chelsea）
[7] ［7］Grigorenko I, Grigorenko E 2003 Phys. Rev. Lett. 91 034101
[8] ［8］Gao X, Yu J B 2005 Chaos, Solitons Fract. 24 1097
[9] ［9］Li C G, Chen G 2004 Chaos, Solitons Fract. 22 549
[10] ］Li C P, Peng G J 2004 Chaos, Solitons Fract. 22 443
[11] ］Gao X, Yu J B 2005 Chin. Phys. 14 908
[12] ］Wang F Q, Liu C X 2006 Acta Phys. Sin. 55 3922 （in Chinese）［王发强，刘崇新 2006 55 3922］
[13] ］Liu C X 2007 Acta Phys. Sin. 56 6865 （in Chinese）［刘崇新 2007 56 6865］
[14] ］Chen X R, Liu C X, Wang F Q, Li Y X 2008 Acta Phys. Sin. 57 1416 （in Chinese）［陈向荣、刘崇新、王发强、李永勋2008 57 1416］
[15] ］Zhang R X, Yang S P 2009 Acta Phys. Sin. 58 2957 （in Chinese）［张若洵、杨世平2009 58 2957］
[16] ］Leipnik R B, Newton T A 1981 Phys. Lett. A 86 63
[17] ］Caputo M 1967 Goephys. J. R. Atr. Soc. 13 529
[18] ］Matignon D 1996 Computational Eng. in Sys. Appl. 12 963
[19] ］Feigenbaum M J 1978 J. Stat. Phys. 19 25
[20] ］Alexander J 1988 Dynamical systems （New York: Springer）
[21] ］Kan I, Kocak H, Yorke J 1992 Annals Math. 136 219
[22] ］Dafilis M P, Bourke P D, Liley D T J, Cadusch P J 2002 Comput. Graph. 26 971
[23] ］Wang X Y, Luo C 2006 Appl. Math. Comput. 183 30
[24] ］Wang X Y 2003 Chaos in the Complex Nonlinearity System （Bejjing: Electronics Industry Press） （in Chinese）［王兴元2003 复杂非线性系统中的混沌（北京：电子工业出版社）
[25] ］Kuruvilla T, Nandakumaran V M 1999 Phys. Lett. A 254 59
[26] ］Sheu L J, Chen H K, Chen J H, Tam L M, Chen W C, Lin K T, Kang Y 2008 Chaos, Solitons Fract. 36 98
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