Imperfect Bifurcation of Systems With Slowly Varying Parameters and Application to Duffing’s Equation
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摘要: 提出新的方法从本质上研究时变参数系统的非完全分岔问题。通过建立时变参数系统的解的线性近似定理去分析时变分岔方程运动的分岔转迁滞后和跃迁现象。利用V函数预测分岔转迁值,将新方法应用于Duffing方程,获得一些新的分岔结果和关于解对初值和参数的敏感性结论。Abstract: A new method a proposed for essentially studying the imperfect bifurcation problem of nonlinear systems with a slowly varying parameter.By establishing some theorems on the solution approximated by that of the linearized system,the delayed bifurcation transition and jump phenomena of the time-dependent equation were analyzed.V-function was used to predict the bifurcation transition value.Applying the new method to analyze the Duffing's equation,some new results about bifurcation as well as that about the sensitivity of the solutions with respect to initial values and parameters are obtained.
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[1] Harberman R. Slowly-varying jump and transition phenomena associated with algebraic bifurcation problems[J]. SIAM J Appl Math,1979,37(1):69-106. [2] Virgin L N. Parametric studies of the dynamic evolution through a fold[J]. J Sound and Vibration,1986,110(1):99-109. [3] Maree G J M. Slow passage through a pitchfork bifurcation[J]. SIAM Appl Math,1996,56(3):889-918. [4] Maree G J M. Sudden exchange in a second-order nonlinear system with a slowly-varying parameter[J]. Int J Non-Linear Mechanics,1993,28(5):1117-1133. [5] Lebovitz N R, Pesci A I. Dynamic bifurcation in Hamiltonian systems with one degree of freedom[J]. SIAM J Appl Math,1995,55(4):69-100. [6] Mandel P, Erneux T, Laser-Lorenz Equations with a time-dependent parameter[J]. Phys Rey Lett,1984,53(19):1818-1820. [7] Erneux T, Mandel P. Stationary harmonic, and pulsed operations of an optical bistable laser with saturable absorber,Ⅱ[J]. Phys Rev A,1984,30(4):1902-1909. [8] Erneux T, Mandel P. Imperfect bifurcation with a slowly varying control parameter[J]. SIAM J Appl Math,1986,46(1):1-15. [9] Mandel P, Erneux T. The slow passage through a steady bifurcation Delay and memory effects[J]. J Statistical Physics,1987,48(5/6):1059-1071. [10] 化存才,陆启韶. 抽运参数随时间慢变所诱发Laser-Lorenz方程的分岔与光学稳态[J]. 物理学报,1999,48(3):408-415. [11] Wiggins S. Introduction to Applied Nonlinear Dynamical Systems and Chaos[M]. New York: Springer-Verlag,1990. [12] 陆启韶. 常微分方程定性方法与分叉[M]. 北京:北京航空航天大学出版社,1989.
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