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Analysis of bursting phenomena in Chen’s system with periodic excitation

Zhang Xiao-Fang Chen Zhang-Yao Bi Qin-Sheng

Analysis of bursting phenomena in Chen’s system with periodic excitation

Zhang Xiao-Fang, Chen Zhang-Yao, Bi Qin-Sheng
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  • For the periodically excited Chens system, when there exists order gap between the natural frequency of the original system and the excited frequency, dynamical behaviors associated with the two different time scales can be observed. Bifurcations of the system have been presented by considering the variation of the excited term. Fast-slow analysis is employed to explore the evolution of the system with different parameter conditions, which gives different types of bursters such as symmetric fold bursting, symmetric subHopf bursting and symmetric Hopf-homoclinic bursting, as well as the bifurcation mechanism. Furthermore, the influence of both the amplitude and the frequency of the excitation on the bursting is discussed in detail.
    • Funds:
    [1]

    [1]Bi Q S 2007 Phys. Lett. A 369 418

    [2]

    [2]Huang J 2008 Nonlin. Anal.: Theory Meth. Appl. 69 4174

    [3]

    [3]Chang J F, Hung M L, Yang Y S, Liao T L, Yan J J 2008 Chaos Solitons Fract. 37 609

    [4]

    [4]Jose A R, Julio S D, Hector P 2005 Phys. Lett. A 338 128

    [5]

    [5]Cai G L, Tan Z M, Zhou W H, Tu W T 2007 Acta Phys. Sin. 56 6230 (in Chinese) [蔡国梁、 谭振梅、 周维怀、 涂文桃 2007 物理学报 56 6230]

    [6]

    [6]Chen G, Ueta T 1999 Int. J. Bifur. Chaos 9 1465

    [7]

    [7]Wang Y W, Guan Z H, Wang H O 2003 Phys. Lett. A 312 34

    [8]

    [8]Chowdhury M S H, Hashim I 2009 Nonlin. Anal. Real World Appl. 10 381

    [9]

    [9]Plienpanich T, Niamsup P, Lenbury Y 2005 Appl. Math. Comput. 171 927

    [10]

    ]Li S H, Cai H X 2004 Acta Phys. Sin. 53 1687 (in Chinese) [李世华、 蔡海兴 2004 物理学报 53 1687]

    [11]

    ]Chen L, Wang D S 2007 Acta Phys. Sin. 56 5661 (in Chinese) [谌龙、 王德石 2007 物理学报 56 5661]

    [12]

    ]Yang Z Q, Lu Q S 2008 Sci. China G 51 687

    [13]

    ]Tanaka H 2006 Phys. Lett. A 350 228

    [14]

    ]Rinzel J 1985 Ordinary and Partial Differential Equations (Berlin: Springer-Verlag) p304

    [15]

    ]Izhikevich E M 2000 Int. J. Bifur. Chaos 10 1171

    [16]

    ]Han X J, Jiang B, Bi Q S 2009 Acta Phys. Sin. 58 4408 ( in Chinese) [韩修静、 江波、 毕勤胜 2009 物理学报 58 4408]

  • [1]

    [1]Bi Q S 2007 Phys. Lett. A 369 418

    [2]

    [2]Huang J 2008 Nonlin. Anal.: Theory Meth. Appl. 69 4174

    [3]

    [3]Chang J F, Hung M L, Yang Y S, Liao T L, Yan J J 2008 Chaos Solitons Fract. 37 609

    [4]

    [4]Jose A R, Julio S D, Hector P 2005 Phys. Lett. A 338 128

    [5]

    [5]Cai G L, Tan Z M, Zhou W H, Tu W T 2007 Acta Phys. Sin. 56 6230 (in Chinese) [蔡国梁、 谭振梅、 周维怀、 涂文桃 2007 物理学报 56 6230]

    [6]

    [6]Chen G, Ueta T 1999 Int. J. Bifur. Chaos 9 1465

    [7]

    [7]Wang Y W, Guan Z H, Wang H O 2003 Phys. Lett. A 312 34

    [8]

    [8]Chowdhury M S H, Hashim I 2009 Nonlin. Anal. Real World Appl. 10 381

    [9]

    [9]Plienpanich T, Niamsup P, Lenbury Y 2005 Appl. Math. Comput. 171 927

    [10]

    ]Li S H, Cai H X 2004 Acta Phys. Sin. 53 1687 (in Chinese) [李世华、 蔡海兴 2004 物理学报 53 1687]

    [11]

    ]Chen L, Wang D S 2007 Acta Phys. Sin. 56 5661 (in Chinese) [谌龙、 王德石 2007 物理学报 56 5661]

    [12]

    ]Yang Z Q, Lu Q S 2008 Sci. China G 51 687

    [13]

    ]Tanaka H 2006 Phys. Lett. A 350 228

    [14]

    ]Rinzel J 1985 Ordinary and Partial Differential Equations (Berlin: Springer-Verlag) p304

    [15]

    ]Izhikevich E M 2000 Int. J. Bifur. Chaos 10 1171

    [16]

    ]Han X J, Jiang B, Bi Q S 2009 Acta Phys. Sin. 58 4408 ( in Chinese) [韩修静、 江波、 毕勤胜 2009 物理学报 58 4408]

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  • Received Date:  17 September 2009
  • Accepted Date:  24 December 2009
  • Published Online:  15 June 2010

Analysis of bursting phenomena in Chen’s system with periodic excitation

  • 1. 江苏大学理学院,镇江 212013

Abstract: For the periodically excited Chens system, when there exists order gap between the natural frequency of the original system and the excited frequency, dynamical behaviors associated with the two different time scales can be observed. Bifurcations of the system have been presented by considering the variation of the excited term. Fast-slow analysis is employed to explore the evolution of the system with different parameter conditions, which gives different types of bursters such as symmetric fold bursting, symmetric subHopf bursting and symmetric Hopf-homoclinic bursting, as well as the bifurcation mechanism. Furthermore, the influence of both the amplitude and the frequency of the excitation on the bursting is discussed in detail.

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