基于Chebyshev正交多项式逼近理论的随机Hopf分岔的研究

发布时间：2018-01-12 17:09

本文关键词：基于Chebyshev正交多项式逼近理论的随机Hopf分岔的研究　出处：《兰州交通大学》2016年硕士论文　论文类型：学位论文

【摘要】：在过去的几十年里,随机分岔是动力学领域中的一个热门话题。基于随机结构和随机动力系统理论,借助切比雪夫多项式逼近法探索非线性随机动力系统的响应,分岔和混沌现象。本文主要对含有随机参数的随机非线性动力系统的Hopf分岔进行了研究。主要内容如下:首先,我们构造了一个含有随机参数的新的二维混沌系统,一个含有随机参数的电机系统和一个含有随机参数的金融系统。通过选择适当的分岔参数分析了系统在平衡点处的稳定性,存在性和发生Hopf分岔的条件。更准确地说,在接下来要研究的系统中我们分别选择参数a,b为分岔参数,当分岔参数a,b穿过临界值0a,0b时,系统就发生Hopf分岔。为了研究这类系统的动力学行为,我们首先借助切比雪夫多项式逼近法将其转换成等价的确定性系统。然后通过第一Lyapunov系数法获得确保这类系统发生Hopf分岔的参数条件。在数值计算过程中借助Maple,Matlab等数学软件得到转化后的高维确定性系统发生Hopf分岔的一些重要结论。并且分析了系统是发生超临界Hopf分岔还是亚临界Hopf分岔,以及发生超临界Hopf分岔时如满足一定的条件,系统从一个不稳定状态变成一个稳定状态。我们可以根据需要去适当的改变系统的参数来避免剧烈波动并且可以解释和预测一些实际问题。其次,借助确定性系统理论对随机系统进行研究,发现其除了具有与确定性系统相似的一些特征外,还表现出一些随机系统特有的特征。与确定性系统不同,随机Hopf分岔临界值的确定不仅取决于随机系统中的随机参数,而且与随机参数的强度有关。当随机参数的强度改变时,随机Hopf分岔的临界值也会随之发生一定的变化。最后,数值模拟的结果证明了本文理论结果是正确有效的。显然,关于这类系统还存在更多有趣的问题比如复杂性,控制,和同步,这些都值得进一步去研究。
[Abstract]:In the past few decades, stochastic bifurcation has been a hot topic in the field of dynamics, based on stochastic structure and stochastic dynamical system theory. The response of nonlinear stochastic dynamical systems is investigated by Chebyshev polynomial approximation. Bifurcation and chaos phenomena. This paper mainly studies the Hopf bifurcation of stochastic nonlinear dynamical systems with random parameters. The main contents are as follows: first. We construct a new two-dimensional chaotic system with random parameters. A motor system with random parameters and a financial system with random parameters are used to analyze the stability of the system at the equilibrium point by selecting proper bifurcation parameters. More accurately, in the system to be studied, we select the parameter ab as the bifurcation parameter respectively, when the bifurcation parameter ab passes through the critical value of 0 a / 0 b. In order to study the dynamic behavior of the system, Hopf bifurcation occurs in the system. We first convert it to an equivalent deterministic system by means of the Chebyshev polynomial approximation method. Then we obtain the parameter conditions by the first Lyapunov coefficient method to ensure the Hopf bifurcation of this kind of system. In the course of numerical calculation, Maple is used. Some important conclusions on the Hopf bifurcation of the transformed high-dimensional deterministic system are obtained by using Matlab and other mathematical software. The supercritical Hopf bifurcation or subcritical Hopf bifurcation is analyzed. . And if the supercritical Hopf bifurcation occurs, some conditions are satisfied. System from an unstable state to a stable state. We can appropriately change the system parameters as necessary to avoid violent fluctuations and can explain and predict some practical problems. Secondly. Based on the theory of deterministic system, it is found that the stochastic system not only has some characteristics similar to the deterministic system, but also shows some unique characteristics of the stochastic system, which is different from the deterministic system. The determination of the critical value of stochastic Hopf bifurcation depends not only on the random parameters in the stochastic system, but also on the strength of the random parameters. The critical value of stochastic Hopf bifurcation will also change with it. Finally, the numerical simulation results show that the theoretical results in this paper are correct and effective. There are many more interesting questions about such systems, such as complexity, control, and synchronization, which deserve further study.
【学位授予单位】：兰州交通大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O175

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