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The complex dynamics of the Logistic map via two types of periodic impulsive forces is investigated in this paper. With the parameter varying, the system produces the phenomenon such as equilibrium solutions, periodic solutions, and chaotic solutions. Furthermore the system can evolve into chaos by a cascading of period-doubling bifurcations. The Poincaré map of the Logistic map via periodic impulsive force is constructed and its bifurcation is analyzed. Finally, the Floquet theory is used to explore the bifurcation mechanism for the periodic solutions.
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Keywords:
- Logistic map /
- impulse /
- periodic solutions /
- bifurcation mechanism
[1] May R M 1976 Nature 261 459
[2] Singh N, Sinha A 2010 Opt. Lasers Eng. 48 398
[3] Stein R R, Isambert H 2011 Phys. Rev. E 84 051904
[4] Nagatani T, Sugiyama N 2013 Physica A 392 851
[5] Jiang H B, Yu J J, Zhou C G 2008 IET Control Theory Appl. 2 654
[6] Qian L N, Lu Q S, Meng Q G, Feng Z S 2010 J. Math. Anal. Appl. 363 345
[7] Zhang L P, Jiang H B, Bi Q S 2010 Chin. Phys. B 19 010507
[8] Wang L, Zhao R, Xu W, Zhang Y 2011 Chin. Phys. B 20 020506
[9] Wang X Y, Zhang Y L, Lin D, Zhang N 2011 Chin. Phys. B 20 030506
[10] Zhou J, Wu Q J, Xiang L 2012 Nonlinear Dyn. 69 1393
[11] Jin L, Lu Q S, Wang Q 2004 Chin. J. Appl. Mech. 21 21 (in Chinese) [金俐, 陆启韶, 王琪 2004 应用力学学报 21 21]
[12] Lu Q S, Jin L 2005 Acta Mech. Sol. Sin. 26 132 (in Chinese) [陆启韶, 金俐 2005 固体力学学报 26 132]
[13] Lenci S, Rega G 2000 Chaos, Solitons and Fractals 11 2453
[14] Jiang G R, Yang Q G 2008 Chin. Phys. B 17 4114
[15] Jiang G R, Xu B G, Yang Q G 2009 Chin. Phys. B 18 5235
[16] Zhang S W, Chen L S 2005 Chaos, Solitons and Fractals 24 73
[17] Georgescu P, Zhang H, Chen L S 2008 Appl. Math. Comput. 202 675
[18] Jiang H B, Zhang L P, Chen Z Y, Bi Q S 2012 Acta Phys. Sin. 61 080505 (in Chinese) [姜海波, 张丽萍, 陈章耀, 毕勤胜 2012 61 080505]
[19] Gao S J, Chen L S 2005 Chaos, Solitons and Fractals 23 519
[20] Liu F 2008 Ph. D. Dissertation (Wuhan: Huazhong University of Science and Technology) (in Chinese) [刘峰 2008 博士学位论文 (武汉: 华中科技大学)]
[21] Liu F, Guan Z H, Wang H O 2010 Nonlinear Anal. Real World Appl. 11 1491
[22] Kawakami H 1984 IEEE Trans. Circuits Syst. 31 248
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[1] May R M 1976 Nature 261 459
[2] Singh N, Sinha A 2010 Opt. Lasers Eng. 48 398
[3] Stein R R, Isambert H 2011 Phys. Rev. E 84 051904
[4] Nagatani T, Sugiyama N 2013 Physica A 392 851
[5] Jiang H B, Yu J J, Zhou C G 2008 IET Control Theory Appl. 2 654
[6] Qian L N, Lu Q S, Meng Q G, Feng Z S 2010 J. Math. Anal. Appl. 363 345
[7] Zhang L P, Jiang H B, Bi Q S 2010 Chin. Phys. B 19 010507
[8] Wang L, Zhao R, Xu W, Zhang Y 2011 Chin. Phys. B 20 020506
[9] Wang X Y, Zhang Y L, Lin D, Zhang N 2011 Chin. Phys. B 20 030506
[10] Zhou J, Wu Q J, Xiang L 2012 Nonlinear Dyn. 69 1393
[11] Jin L, Lu Q S, Wang Q 2004 Chin. J. Appl. Mech. 21 21 (in Chinese) [金俐, 陆启韶, 王琪 2004 应用力学学报 21 21]
[12] Lu Q S, Jin L 2005 Acta Mech. Sol. Sin. 26 132 (in Chinese) [陆启韶, 金俐 2005 固体力学学报 26 132]
[13] Lenci S, Rega G 2000 Chaos, Solitons and Fractals 11 2453
[14] Jiang G R, Yang Q G 2008 Chin. Phys. B 17 4114
[15] Jiang G R, Xu B G, Yang Q G 2009 Chin. Phys. B 18 5235
[16] Zhang S W, Chen L S 2005 Chaos, Solitons and Fractals 24 73
[17] Georgescu P, Zhang H, Chen L S 2008 Appl. Math. Comput. 202 675
[18] Jiang H B, Zhang L P, Chen Z Y, Bi Q S 2012 Acta Phys. Sin. 61 080505 (in Chinese) [姜海波, 张丽萍, 陈章耀, 毕勤胜 2012 61 080505]
[19] Gao S J, Chen L S 2005 Chaos, Solitons and Fractals 23 519
[20] Liu F 2008 Ph. D. Dissertation (Wuhan: Huazhong University of Science and Technology) (in Chinese) [刘峰 2008 博士学位论文 (武汉: 华中科技大学)]
[21] Liu F, Guan Z H, Wang H O 2010 Nonlinear Anal. Real World Appl. 11 1491
[22] Kawakami H 1984 IEEE Trans. Circuits Syst. 31 248
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