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具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

王万永 陈丽娟

王万永, 陈丽娟. 具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔[J]. 应用数学和力学, 2013, 34(7): 764-770. doi: 10.3879/j.issn.1000-0887.2013.07.012
引用本文: 王万永, 陈丽娟. 具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔[J]. 应用数学和力学, 2013, 34(7): 764-770. doi: 10.3879/j.issn.1000-0887.2013.07.012
WANG Wan-yong, CHEN Li-juan. Weak Resonant Double Hopf Bifurcation of n van der Pol Oscillators With Delay Coupling[J]. Applied Mathematics and Mechanics, 2013, 34(7): 764-770. doi: 10.3879/j.issn.1000-0887.2013.07.012
Citation: WANG Wan-yong, CHEN Li-juan. Weak Resonant Double Hopf Bifurcation of n van der Pol Oscillators With Delay Coupling[J]. Applied Mathematics and Mechanics, 2013, 34(7): 764-770. doi: 10.3879/j.issn.1000-0887.2013.07.012

具有时滞耦合的n个van der Pol振子弱共振双Hopf分岔

doi: 10.3879/j.issn.1000-0887.2013.07.012
详细信息
    作者简介:

    王万永(1982—),男,河南南阳人,讲师,博士(通讯作者. E-mail:wangwanyong630@163.com).

  • 中图分类号: O322;O175.1

Weak Resonant Double Hopf Bifurcation of n van der Pol Oscillators With Delay Coupling

  • 摘要: 研究了具有时滞耦合的n个van der Pol振子系统中发生的弱共振双Hopf分岔.应用改进的多尺度方法,得到了2∶5共振的复振幅方程.通过将复振幅设为极坐标形式,将复振幅方程转化为一个二维的实振幅系统.通过研究实振幅方程的平衡点及其稳定性,对系统在2∶5共振点附近的动力学行为进行了开折和分类.得到了一些有趣的动力学现象,如振幅死区、周期解和双稳态解等,相应的数值模拟验证了理论结果的正确性.
  • [1] Brailove A A, Linsay P S, Koster G. An experimental study of a population of relaxation oscillators with a phase-repelling mean-field coupling[J]. International Journal of Bifurcation Chaos,1996, 6(2): 1211-1253.
    [2] Ramana Reddy D V, Sen A, Johnston G L. Experimental evidence of time delay induced death in coupled limit cycle oscillators[J].Physical Review Letters,2000, 85(16): 3381-3384.
    [3] Satoh K. Computer experiments on the co-operative behavior of a network of interacting nonlinear oscillators[J].Journal of Physical Society of Japan,1989, 58: 2010-2021.
    [4] Hadley P, Beasley M R, Wiesenfeld K. Phase locking of Josephson-junction series arrays[J].Physical Review B,1988, 38(13): 8712-8719.
    [5] Nakajima K, Sawada Y. Experimental studies on the weak coupling of oscillatory chemical reaction systems[J].Journal of Chemical Physics,1980, 72(4): 2231-2234.
    [6] Bar-Eli K. On the stability of coupled chemical oscillators[J].Physica D,1985, 14(2): 242-252.
    [7] Shiino M, Frankowicz M. Synchronization of infinitely many coupled limit-cycle oscillators[J].Physical Letter A,1989, 136(3): 103-108.
    [8] Aronson D G, Ermentrout G B, Koppel N. Amplitude response of coupled oscillators[J].Physica D,1990, 41(3): 403-449.
    [9] Mirollo R E, Strogatz S H. Amplitudes death in an array of limit-cycle oscillators[J].Journal of Statistical Physics,1990, 60(1/2): 245-262.
    [10] Collins J J, Stewart I N. Coupled nonlinear oscillators and the symmetries of animal gaits[J].Journal of Nonlinear Science,1993, 3(1): 349-392.
    [11] Daido H. Onset of cooperative entrainment in limit-cycle oscillators with uniform all-to-all interactions: bifurcation of the order function[J].Physica D,1996, 91(1/2): 24-66.
    [12] Pecora L M. Synchronization conditions and desynchronizing patterns in coupled limit-cycle and chaotic systems[J].Physical Review E,1998, 58(1): 347-360.
    [13] Zhang C R, Zheng B D, Wang L C. Multiple Hopf bifurcation of three coupled van der Pol oscillators with delay[J].Applied Mathematics and Computation,2011, 217(1): 7155-7166.
    [14] Song Y L, Xu J, Zhang T H. Bifurcation, amplitude death and oscillation patterns in a system of three coupled van der Pol oscillators with diffusively delayed velocity coupling[J].Chaos,2011, 21(2):023111.
    [15] Barrón M A, Sen M. Synchronization of four coupled van der Pol oscillators[J].Nonlinear Dynamics,2009, 56(4):357-367.
    [16] Hirano N, Rybicki S. Existence of limit cycles for coupled van der Pol equations[J].Journal of Differential Equations,2003, 195(1):194-209.
    [17] Kovacic I, Mickens R E. A generalized van der Pol type oscillator: investigation of the properties of its limit cycle[J].Mathematical and Computer Modelling,2012, 55(9): 645-653.
    [18] Zhang C M, Li W X, Wang K. Boundedness for network of stochastic coupled van der Pol oscillators with time-varying delayed coupling[J].Applied Mathematical Modelling, 2013, 37(7): 5394-5402.
    [19] Xiao M, Zheng W X, Cao J D. Approximate expressions of a fractional order van der Pol oscillator by the residue harmonic balance method[J].Mathematics and Computers in Simulation, 2013, 89(1): 1-12.
    [20] Niebur E, Schuster H G, Kammen D. Collective frequencies and metastability in networks of limit-cycle oscillators with time delay[J].Physical Review Letters,1991, 67(20): 2753-2756.
    [21] Nakamura Y, Tominaga F, Munakata T. Clustering behavior of time-delayed nearest-neighbor coupled oscillators[J].Physical Review E,1994, 49(6): 4849-4856.
    [22] Seunghwan K, Seon H, Ryu C S. Multistability in coupled oscillator systems with time delay[J].Physical Review Letters,1997, 79(15): 2911-2914.
    [23] Wang W Y, Xu J. Multiple scales analysis for double Hopf bifurcation with 1∶3 resonance[J].Nonlinear Dynamics,2011, 66(1/2): 39-51.
    [24] Luongo A, Paolone A, Di Egidio A. Multiple timescales analysis for 1∶2 and 1∶3 resonant Hopf bifurcations[J].Nonlinear Dynamics,2003, 34(3/4)∶ 269-291.
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出版历程
  • 收稿日期:  2013-04-07
  • 修回日期:  2013-05-25
  • 刊出日期:  2013-07-15

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