Zhang Wei, Huo Quan-zhong, Li Li. Heteroclinic Orbit and Subharmonic Bifurcations and Chaos of Nonlinear Oscillator[J]. Applied Mathematics and Mechanics, 1992, 13(3): 199-208.
Citation: Zhang Wei, Huo Quan-zhong, Li Li. Heteroclinic Orbit and Subharmonic Bifurcations and Chaos of Nonlinear Oscillator[J]. Applied Mathematics and Mechanics, 1992, 13(3): 199-208.

Heteroclinic Orbit and Subharmonic Bifurcations and Chaos of Nonlinear Oscillator

  • Received Date: 1991-01-21
  • Publish Date: 1992-03-15
  • Dynamical behavior of nonlinear oscillator under combined parametric and forcing excitation, which includes yon der Pol damping, is very complex. In this paper, Melnikov's method is used to study the heteroclinic orbit bifurcations, subharmonic bifurcations and chaos in this system. Smale horseshoes and chaotic motions can occur from odd subharmonic bifurcation of infinite order in this system-far various resonant cases finally the numerical computing method is used to study chaotic motions of this system. The results achieved reveal some new phenomena.
  • loading
  • [1]
    Holmes, P. J. and R. A. Rand, Phase portraits and bifurcations of the nonlinear oscillator:x+(a+γx2)x-βx+δx3=0 Int. J. Nonlinear Mech., 15, 1 (1980), 449-458.
    [2]
    Greenspan B. D. and P. J. Holmes, Repeated resonance and homoclinic bifurcation in a periodically forced family of oscillators, SIAM J. Math. Anal., 15 (1984), 69-97.
    [3]
    唐建宁、刘曾荣,2-jet和3-jet 系统中的复杂分叉现象,应用数学学报,11(2) (1988), 173-181
    [4]
    Guckenheimer, J. and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York (1983).
    [5]
    Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc., 73 (1967), 747-817.
    [6]
    Greenspan, B. D., and P. J. Holems, Homoclinic orbits, subharmonics and global bifurcations in forced oscillations, Nonlinear Dynamics and Turbulence, G. Barenblatt, G. Ioose, and D. D. Joseph(eds), Pitman, London, (1983), 172-214.
    [7]
    Melnikov, V. K., On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc., 12 (1963), 1-57.
    [8]
    Holmes, P. J., Averaging and chaotic motions in forced oscillations, SIAM J. Appl. Math., 38(1980), 65-80.
    [9]
    Hale, J. K., Ordinary Differential Equations, 2nd Edition, Kreiger Publ. Co. (1980).
    [10]
    Hale, J. K. and X.-B. Lin, Heteroclinic orbits for retarded functional differential equation,J. Diff. Eqs., 65 (1986), 175-202.
    [11]
    Gradshteyn, I. S. and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press (1980).
    [12]
    万世栋、李继彬,Jacobi椭圆函数有理式的Fourier级数.应用数学和力学,9 (6) (1988),499-513
    [13]
    Brunsden, V., J. Cortell and P. J. Holmes, Power spectra of chaotic vibrations of a buckled beam, J. Sound Vib., 130, 1(1989), 1-25.
  • 加载中

Catalog

    通讯作者: 陈斌, bchen63@163.com
    • 1. 

      沈阳化工大学材料科学与工程学院 沈阳 110142

    1. 本站搜索
    2. 百度学术搜索
    3. 万方数据库搜索
    4. CNKI搜索

    Article Metrics

    Article views (2311) PDF downloads(770) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return