CHEN Yang-yang, YAN Le-wei, SZE Kam-yim, CHEN Shu-hui. Generalized Hyperbolic Perturbation Method for Homoclinic Solutions of Strongly Nonlinear Autonomous Systems[J]. Applied Mathematics and Mechanics, 2012, 33(9): 1064-1077. doi: 10.3879/j.issn.1000-0887.2012.09.004
Citation: CHEN Yang-yang, YAN Le-wei, SZE Kam-yim, CHEN Shu-hui. Generalized Hyperbolic Perturbation Method for Homoclinic Solutions of Strongly Nonlinear Autonomous Systems[J]. Applied Mathematics and Mechanics, 2012, 33(9): 1064-1077. doi: 10.3879/j.issn.1000-0887.2012.09.004

Generalized Hyperbolic Perturbation Method for Homoclinic Solutions of Strongly Nonlinear Autonomous Systems

doi: 10.3879/j.issn.1000-0887.2012.09.004
  • Received Date: 2012-05-08
  • Rev Recd Date: 2012-05-16
  • Publish Date: 2012-09-15
  • A generalized hyperbolic perturbation method was presented for homoclinic solutions of strongly nonlinear autonomous oscillators, in which the perturbation procedure was improved for those systems whose exact homoclinic generating solutions could not be explicitly derived. The generalized hyperbolic functions were employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method. Several strongly nonlinear oscillators with quadratic, cubic and quartic nonlinearity were studied in details to illustrate the efficiency and accuracy of the present method.
  • loading
  • [1]
    陈树辉. 强非线性振动系统的定量分析方法[M]. 北京:科学出版社, 2007. (CHEN Shu-hui. Quantitative Analysis Methods for Strongly Nonlinear Vibration[M]. Beijing: Science Press, 2007. (in Chinese))
    [2]
    刘曾荣. 混沌研究中的解析方法[M]. 上海:科技教育出版社, 2002. (LIU Zeng-rong. Analytical Methods for Study of Chaos[M]. Shanghai: Science and Technology Education Press, 2002. (in Chinese))
    [3]
    Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[M]. New York: Springer, 1983.
    [4]
    Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M]. New York: Springer, 1990.
    [5]
    Nayfeh A H, Balachandran B. Applied Nonlinear Dynamics, Analytical, Computational, and Experimental Methods[M]. New York: Wiley, 1995.
    [6]
    Li J B, Dai H H. On the Study of Singular Nonlinear Traveling Wave Equation: Dynamical System Approach[M]. Beijing: Science Press, 2005.
    [7]
    陈予恕, 丁千. C-L 方法及其在工程非线性动力学问题中的应用[J]. 应用数学和力学, 2001, 22(2) : 127-134. (CHEN Yu-shu, DING Qian. C-L method and its application to engineering nonlinear dynamical problems[J]. Applied Mathematics and Mechanics(English Edition), 2001, 22(2) : 144-153.)
    [8]
    陈立群. 带慢变角参数摄动平面非Hamilton可积系统的混沌[J]. 应用数学和力学, 2001, 22(11): 1172-1176. (CHEN Li-qun. Chaos in pertrubation planar non-hamiltonian integrable systems with slowly-varying angle parameters[J]. Applied Mathematics and Mechanics(English Edition), 2001, 22(11) : 1301-1305.)
    [9]
    Vakakis A F. Exponentially small splittings of manifolds in a rapidly forced Duffing system, Letter to the editor[J]. Journal of Sound and Vibration, 1994, 170(1): 119-129.
    [10]
    Vakakis A F, Azeez M F A. Analytic approximation of the homoclinic orbits of the Lorenz system at σ= 10, b= 8/3 and ρ=13.926…[J]. Nonlinear Dynamics, 1998, 15(3): 245-257.
    [11]
    Xu Z, Chan H S Y, Chung K W. Separatrices and limit cycles of strongly nonlinear oscillators by the perturbation-incremental method[J]. Nonlinear Dynamics, 1996, 11(3): 213-233.
    [12]
    Chan H S Y, Chung K W, Xu Z. Stability and bifurcations of limit cycles by the perturbation-incremental method[J]. Journal of Sound and Vibration, 1997, 206(4): 589-604.
    [13]
    Belhaq M. Predicting homoclinic bifurcations in planar autonomous systems[J]. Nonlinear Dynamics, 1999, 18(4): 303-310.
    [14]
    Belhaq M, Lakrad F. Prediction of homoclinic bifurcation: the elliptic averaging method[J]. Chaos Solitons & Fractals, 2000, 11(14): 2251-2258.
    [15]
    Belhaq M, Fiedler B, Lakrad F. Homoclinic connections in strongly self-excited nonlinear oscillators: the Melnikov function and the elliptic Lindstedt-Poincaré method[J]. Nonlinear Dynamics, 2000, 23(1): 67-86.
    [16]
    Mikhlin Y V. Analytical construction of homoclinic orbits of two- and three-dimensional dynamical systems[J]. Journal of Sound and Vibration, 2000, 230(5): 971-983.
    [17]
    Mikhlin Y V, Manucharyan G V. Construction of homoclinic and heteroclinic trajectories in mechanical systems with several equilibrium positions[J]. Chaos, Solitons & Fractals, 2003, 16(2): 299-309.
    [18]
    Manucharyan G V, Mikhlin Y V. The construction of homo- and heteroclinic orbits in non-linear systems[J]. Journal of Applied Mathematics and Mechanics, 2005, 69(1): 39-48.
    [19]
    Cao H J, Jiang Y Z, Shan Y L. Primary resonant optimal control for nested homoclinic and heteroclinic bifurcations in single-dof nonlinear oscillators[J]. Journal of Sound and Vibration, 2006, 289(1/2): 229-244.
    [20]
    Zhang Q C, Wang W, Li W Y. Heteroclinic bifurcations of strongly nonlinear oscillator[J]. Chinese Physics Letters, 2008, 25(5): 1905-1907.
    [21]
    Zhang Y M, Lu Q S. Homoclinic bifurcation of strongly nonlinear oscillators by frequency-incremental method[J]. Communications in Nonlinear Science and Numerical Simulation, 2003, 8(1): 1-7.
    [22]
    Izydorek M, Janczewska J. Homoclinic solutions for a class of the second order Hamiltonian systems[J]. Journal of Differential Equations, 2005, 219(2): 375-389.
    [23]
    Izydorek M, Janczewska J. Heteroclinic solutions for a class of the second order Hamiltioinian systems[J]. Journal of Differential Equations, 2007, 238(2): 381-393.
    [24]
    Cao Y Y, Chung K W, Xu J. A novel construction of homoclinic and heteroclinic orbits in nonlinear oscillators by a perturbation-incremental method[J]. Nonlinear Dynamics, 2011, 64(3): 221-236.
    [25]
    Chen S H, Chen Y Y, Sze K Y. A hyperbolic perturbation method for determining homoclinic solution of certain strongly nonlinear autonomous oscillators[J]. Journal of Sound and Vibration, 2009, 322(1/2): 381-392.
    [26]
    Chen Y Y, Chen S H. Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by the hyperbolic perturbation method[J]. Nonlinear Dynamics, 2009, 58(1/2): 417-429.
    [27]
    Chen Y Y, Chen S H, Sze K Y. A hyperbolic Lindstedt-Poincaré method for homoclinic motion of a kind strongly nonlinear autonomous oscillators[J]. Acta Mechanica Sinica, 2009, 25(5): 721-729.
    [28]
    Chen S H, Chen Y Y, Sze K Y. Homoclinic and heteroclinic solutions of cubic strongly nonlinear autonomous oscillators by hyperbolic Lindstedt-Poincaré method[J]. Science China, Technological Science, 2010, 53(3): 1-11.
    [29]
    Abramowitz M, Stegun I A. Handbook of Mathematical Functions[M]. New York: Dover, 1972.
    [30]
    Merkin J H, Needham D J. On infinite period bifurcations with an application to roll waves[J]. Acta Mechanica, 1986, 60(1/2): 1-16.
  • 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 (1708) PDF downloads(949) Cited by()
    Proportional views
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return