周期脉冲作用下Logistic映射的复杂动力学行为及其分岔分析

姜海波; 李涛; 曾小亮; 张丽萍

doi:10.7498/aps.62.120508

摘要

研究了两种周期脉冲作用下Logistic映射的复杂动力学行为. 随着参数的变化, 该系统产生平衡解、周期解、混沌等现象, 且该系统可经级联倍周期分岔到达混沌. 通过构造Poincaré 映射, 对周期脉冲作用下Logistic映射进行了分岔分析. 最后基于Floquet理论揭示了该系统周期解的分岔机理.

关键词:

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.

Keywords:

作者及机构信息

1.
盐城师范学院数学科学学院, 盐城 224002

基金项目: 国家自然科学基金(批准号: 11202180, 61273106, 11171290)、江苏省自然科学基金(批准号: BK2010292, BK2010293)、江苏省高校自然科学基金(批准号: 10KJB510026)、国家级大学生创新创业训练计划(批准号: 201210324009)和江苏省大学生实践创新训练计划 (批准号: 2012JSSPITP2378)资助的课题.

Authors and contacts

1.
School of Mathematics, Yancheng Teachers University, Yancheng 224002, China

Funds: Project supported by the National Nature Science Foundation of China (Grant Nos. 11202180, 61273106, 11171290), the Natural Science Foundation of Jiangsu Province, China (Grant Nos. BK2010292, BK2010293), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 10KJB510026), the National Training Programs of Innovation and Entrepreneurship for Undergraduates, China (Grant No. 201210324009), and the Training Programs of Practice and Innovation for Jiangsu College Students, China (Grant No. 2012JSSPITP2378).

参考文献

[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 物理学报 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

施引文献

[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 物理学报 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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