随机分岔的正交多项式逼近分析

【作者】 马少娟；

【导师】 徐伟；

【作者基本信息】 西北工业大学 ， 统计学， 2005， 硕士

【摘要】 近年来,由于计算工具的更新和新的数学分析方法的应用,非线性随机动力学理论得到了迅速发展,但是还存在许多领域需要进一步探索。本文应用正交多项式逼近法针对具有随机参数的非线性随机动力学系统的随机分岔问题作了初步研究,主要工作包含以下三个方面: 1.应用Chebyshev多项式逼近法把含有服从拱形分布随机变量的Van der Pol系统转化为等价确定性扩阶系统,再利用一般数值方法研究了该系统的对称破裂分岔和倍周期分岔行为。结果显示此类van der Pol系统有着与确定性系统相似的分岔行为,但是受随机因素影响,外激激励的幅值或频率增大时,系统的分岔点会前移。这一结果在现有文献中还没有见到。 2.文中介绍了具有更普遍意义的概率密度函数模型—λ类概率密度函数,并应用相应的Gegenbauer多项式逼近将具有服从此类分布随机变量的Duffing-Van der Pol系统转化为等价的确定性系统,借此研究了系统的随机分岔行为,数值模拟结果显示当随机参数的强度变大时,系统的随机因素就强烈影响到了其非线性行为。此类系统的这些性质现有文献中还没有见到,本文的研究在实际工程问题中有着重要的实用价值。 3.将正交多项式逼近法推广到具有两个相互独立的随机参数的非线性系统,以Duffing系统为例,数值模拟显示随着两个随机参数强度的增加,Duffing系统提前进入分岔状态,由原来的周期1T变为周期2T,周期2T变为周期4T。进一步的推广证实正交多项式逼近法在实际问题中有广阔的应用前景,它可以解决含有多个随机参数的非线性随机动力学系统的分岔问题。更多还原

【Abstract】 The study of nonlinear stochastic dynamic system is quickly developed through new computational tools and new methods of mathematic analysis, but in this field there are more unknown facets we have not recognized. This thesis is devoted to explore bifurcation problems in nonlinear stochastic dynamics system with random parameter via orthogonal polynomial approximation method. The main works as follows:Firstly, stochastic van der Pol system with random parameter subject to an arch-like probability density function is reduced into its equivalent deterministic one using Chebyshev polynomial approximation method. The numerical results show that similar to their counterpart in deterministic system the symmetry-breaking bifurcation and period-doubling bifurcation occur in the stochastic van der Pol system. But for random factor the point of period-doubling bifurcation can move in the stochastic system as increasing the frequency or amplitude of harmonic excitation.Secondly, the A -probability density function is more commonly applied in engineering and physics is introduced. Then we reduce stochastic Duffing-van der Pol system into its equivalent deterministic system by Gegenbauer polynomial approximation method. The stochastic bifurcations of stochastic system are analyzed by numerical method. Comparing with deterministic system, the point of bifurcation in stochastic Duffing-van der Pol system is moved as increasing the intensity of random parameter. These results are available to practical application.Thirdly, we generalize the orthogonal polynomial method to nonlinear system with two mutual independent random parameters. Taking stochastic Duffing system as an example, the stochastic bifurcations are explored by numerical method. Under effect of the intensity of random parameters, the period-doubling bifurcations in stochastic Duffing system are more complex. From these results we can prove that this method can solve the bifurcation problems of the nonlinear system with several random parameters.更多还原