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依据分数阶线性系统的稳定性理论,研究了具有双重混沌吸引子的Newton-Leipnik系统取不同分数阶时的动力学行为.研究表明该系统具有逆向Hopf分岔过程,即随着阶数的下降,分数阶Newton-Leipnik系统由双重混沌吸引子突变为单吸引子,其动力学行为将由混沌态历经短暂的周期态后收敛于稳定的平衡点.
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关键词:
- Newton-Leipnik系统 /
- 分数阶 /
- 双重吸引子
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.-
Keywords:
- Newton-Leipnik system /
- fractional order /
- double attractors
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[2] [2]Hilfer R 2001 Applications of fractional calculus in physics (New Jersey: World Scientific)
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[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 物理学报 55 3922]
[13] ]Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese)[刘崇新 2007 物理学报 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 物理学报 57 1416]
[15] ]Zhang R X, Yang S P 2009 Acta Phys. Sin. 58 2957 (in Chinese)[张若洵、杨世平2009 物理学报 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 物理学报 55 3922]
[13] ]Liu C X 2007 Acta Phys. Sin. 56 6865 (in Chinese)[刘崇新 2007 物理学报 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 物理学报 57 1416]
[15] ]Zhang R X, Yang S P 2009 Acta Phys. Sin. 58 2957 (in Chinese)[张若洵、杨世平2009 物理学报 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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