Please wait a minute...
Chin. Phys. B, 2014, Vol. 23(1): 010501    DOI: 10.1088/1674-1056/23/1/010501
GENERAL Prev   Next  

Bifurcation analysis of the logistic map via two periodic impulsive forces

Jiang Hai-Bo, Li Tao, Zeng Xiao-Liang, Zhang Li-Ping
School of Mathematics, Yancheng Teachers University, Yancheng 224002, China
Abstract  The complex dynamics of the logistic map via two periodic impulsive forces is investigated in this paper. The influences of the system parameter and the impulsive forces on the dynamics of the system are studied respectively. With the parameter varying, the system produces the phenomenon such as periodic solutions, chaotic solutions, and chaotic crisis. Furthermore, the system can evolve to chaos by a cascading of period-doubling bifurcations. The Poincaré map of the logistic map via two periodic impulsive forces is constructed and its bifurcation is analyzed. Finally, the Floquet theory is extended to explore the bifurcation mechanism for the periodic solutions of this non-smooth map.
Keywords:  logistic map      impulse      periodic solutions      bifurcation mechanism  
Received:  15 May 2013      Revised:  07 June 2013      Published:  12 November 2013
PACS:  05.45.Ac (Low-dimensional chaos)  
  05.45.Pq (Numerical simulations of chaotic systems)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11202180, 61273106, and 11171290), the Natural Science Foundation of Jiangsu Province, China (Grant Nos. BK2010292 and BK2010293), 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).
Corresponding Authors:  Jiang Hai-Bo     E-mail:

Cite this article: 

Jiang Hai-Bo, Li Tao, Zeng Xiao-Liang, Zhang Li-Ping Bifurcation analysis of the logistic map via two periodic impulsive forces 2014 Chin. Phys. B 23 010501

[1] May R M 1976 Nature 261 459
[2] Singh N and Sinha A 2010 Opt. Lasers Eng. 48 398
[3] Stein R R and Isambert H 2011 Phys. Rev. E 84 051904
[4] Nagatani T and Sugiyama N 2013 Physica A 392 851
[5] Bainov D D and Simeonov P S 1989 Systems with Impulse Effect: Stability, Theory and Applications (New York: Halsted Press)
[6] Lakshmikantham V, Bainov D D and Simeonov P S 1989 Theory of Impulsive Differential Equations (Singapore: World Scientific)
[7] Yang T 2001 Impulsive Control Theory (Berlin: Springer)
[8] Jiang H B, Yu J J and Zhou C G 2008 IET Control Theory Appl. 2 654
[9] Zhang L P, Jiang H B and Bi Q S 2010 Chin. Phys. B 19 010507
[10] Qian L N, Lu Q S, Meng Q G and Feng Z S 2010 J. Math. Anal. Appl. 363 345
[11] Wang L, Zhao R, Xu W and Zhang Y 2011 Chin. Phys. B 20 020506
[12] Wang X Y, Zhang Y L, Lin D and Zhang N 2011 Chin. Phys. B 20 030506
[13] Zhou J, Wu Q J and Xiang L 2012 Nonlinear Dyn. 69 1393
[14] Jin L, Lu Q S and Wang Q 2004 Chin. J. Appl. Mech. 21 21
[15] Lu Q S and Jin L 2005 Acta Mech. Solida Sin. 26 132
[16] Lenci S and Rega G 2000 Chaos Soliton. Fract. 11 2453
[17] Jiang G R and Yang Q G 2008 Chin. Phys. B 17 4114
[18] Jiang G R, Xu B G and Yang Q G 2009 Chin. Phys. B 18 5235
[19] Zhang S W and Chen L S 2005 Chaos Soliton. Fract. 24 73
[20] Georgescu P, Zhang H and Chen L S 2008 Appl. Math. Comput. 202 675
[21] Jiang H B, Zhang L P, Chen Z Y and Bi Q S 2012 Acta Phys. Sin. 61 080505 (in Chinese)
[22] Gao S J and Chen L S 2005 Chaos Soliton. Fract. 23 519
[23] Liu F, Guan Z H and Wang H O 2010 Nonlinear Anal. Real World Appl. 11 1491
[24] Jiang H B, Li T, Zeng X L and Zhang L P Acta Phys. Sin. 62 120508 (in Chinese)
[25] Wang W M, Wang X Q and Lin Y Z 2008 Chaos Soliton. Fract. 37 1427
[26] Chen Y P and Liu Z J 2009 Chaos Soliton. Fract. 39 1698
[27] Ma Z J, Yang J and Jiang G R 2010 Appl. Math. Comput. 217 3453
[28] Georgescua P and Zhang H 2012 BioSystems 110 162
[29] Kawakami H 1984 IEEE Trans. Circ. Syst. 31 248
[1] Solid-like ablation propulsion generation in nanosecond pulsed laser interaction with carbon-doped glycerol
Zhi-Yuan Zheng(郑志远), Si-Qi Zhang(张思齐), Tian Liang(梁田), Jing Qi(齐婧), Wei-Chong Tang(汤唯冲), Ke Xiao(肖珂), Lu Gao(高禄), Hua Gao(高华), Zi-Li Zhang(张自力). Chin. Phys. B, 2017, 26(3): 035203.
[2] A self-cited pixel summation based image encryption algorithm
Guo-Dong Ye(叶国栋), Xiao-Ling Huang(黄小玲), Leo Yu Zhang(张愉), Zheng-Xia Wang(王政霞). Chin. Phys. B, 2017, 26(1): 010501.
[3] Bursting phenomena as well as the bifurcation mechanism in a coupled BVP oscillator with periodic excitation
Xiaofang Zhang(张晓芳), Lei Wu(吴磊), Qinsheng Bi(毕勤胜). Chin. Phys. B, 2016, 25(7): 070501.
[4] Complex dynamics analysis of impulsively coupled Duffing oscillators with ring structure
Jiang Hai-Bo, Zhang Li-Ping, Yu Jian-Jiang. Chin. Phys. B, 2015, 24(2): 020502.
[5] A fast image encryption algorithm based on only blocks in cipher text
Wang Xing-Yuan, Wang Qian. Chin. Phys. B, 2014, 23(3): 030503.
[6] Cluster synchronization of uncertain complex networks with desynchronizing impulse
Cai Guo-Liang, Jiang Sheng-Qin, Cai Shui-Ming, Tian Li-Xin. Chin. Phys. B, 2014, 23(12): 120505.
[7] Dynamical investigation and parameter stability region analysis of a flywheel energy storage system in charging mode
Zhang Wei-Ya, Li Yong-Li, Chang Xiao-Yong, Wang Nan. Chin. Phys. B, 2013, 22(9): 098401.
[8] Forced bursting and transition mechanism in CO oxidation with three time scales
Li Xiang-Hong, Bi Qin-Sheng. Chin. Phys. B, 2013, 22(4): 040504.
[9] Bursting oscillation in CO oxidation with small excitation and the enveloping slow-fast analysis method
Li Xiang-Hong, Bi Qin-Sheng. Chin. Phys. B, 2012, 21(6): 060505.
[10] Dynamical behaviors of a system with switches between the Rössler oscillator and Chua circuits
Zhang Chun, Yu Yue, Han Xiu-Jing, Bi Qin-Sheng. Chin. Phys. B, 2012, 21(10): 100501.
[11] The periodic solutions for coupled integrable dispersionless equations
Liu Shi-Kuo, Zhao Qiang, Liu Shi-Da. Chin. Phys. B, 2011, 20(4): 040202.
[12] Symmetric bursting behaviour in non-smooth Chua's circuit
Ji Ying, Bi Qin-Sheng. Chin. Phys. B, 2010, 19(8): 080510.
[13] Wavelet threshold method of resolving noise interference in periodic short-impulse signals chaotic detection
Deng Ke, ZhangLu, Luo Mao-Kang. Chin. Phys. B, 2010, 19(3): 030506.
[14] Cognitive radio resource allocation based on coupled chaotic genetic algorithm
Zu Yun-Xiao, Zhou Jie, Zeng Chang-Chang. Chin. Phys. B, 2010, 19(11): 119501.
[15] Theoretical analysis and numerical simulation of the impulse delivering from laser-produced plasma to solid target
Yang Yan-Nan, Yang Bo, Zhu Jin-Rong, Shen Zhong-Hua, Lu Jian, Ni Xiao-Wu. Chin. Phys. B, 2008, 17(4): 1318-1325.
No Suggested Reading articles found!