【作者】 高琴；

【导师】 黄东卫；

【作者基本信息】 天津工业大学 ， 应用数学， 2006， 硕士

【摘要】 本文简要介绍了非线性动力学和混沌的产生及发展过程,给出了目前对混沌的几种不同的定义、通向混沌的道路及混沌的判定方法。主要介绍了其中一种著名的解析方法——Meinikov方法,它可以研究带有弱周期扰动项的具有同宿轨线或异宿轨线的二阶常微分方程和具有鞍焦型同宿轨线的三阶常微分方程。对于这两类系统,利用一定的技巧,可以建立一个二维庞加莱(Poincare)映射。考虑系统的Poincare映射的鞍点稳定流形与不稳定流形的距离,并用正比于此距离的一个积分——Melnikov函数来判断系统是否出现横截同宿点和横截异宿点,从而判断系统中是否存在混沌运动。 在实际问题的很多情况下,随机扰动是不可避免的,也是不可忽略的,所以,对随机激励的研究也越来越重要。这里我们研究的弱随机外激设为有界噪声,此时,Mdnikov函数变成Meinikov过程,因此,需要从某种概率或统计意义上说随机Melnikov过程是否具有简单零点,这里我们仅从均值及均方意义下考虑,并通过两类Lienard方程及一类Duffing方程来具体说明了Melnikov方法的计算过程,并由此看到了Melnikov方法在随机系统中处理简单零点的困难及计算的复杂性,因此。我们又引入了rate of phase space flux理论,将求解Melnikov函数简单零点的问题转化为求解相流函数的零点,成功避开了计算随机Melnikov函数简单零点的难题,并简化了计算。更多还原

【Abstract】 In this text, we introduce the appearance and development of the nonlinear dynamic system and chaos;give the different definitions of chaos, the routes to chaos and the judgment of chaos. We introduce a famous analytic method——Melnikov method, which can be used to deal with the second order ordinary differential equations with weak period perturbation term and homoclinic orbit or heteroclinic orbit and the three order differential equations with saddle-focus homoclinic orbit. For these two classes of equations, we can establish a two-dimensional Poincare map using some technique. Consider the distance between the stable manifold and the unstable manifold of the saddle of the Poincare map, using an integral called Melnikov function which is direct proportion of the distance to judge if the system will appear transversal homoclinic point and transversal heteroclinic point, and then to judge if the system exist chaotic motion.The stochastic perturbation is unavoidable and cannot be neglected in many practical problems, so the study on stochastic drive is more and more important. Here the stochastic drive we used is bounded noise, then the Melnikov function turns into the Melnikov process, so we need to say if the Melnikov process has simple zero in some probability or statistical sense, here we consider the mean sense and mean square sense only. We also give examples about two classes of Lienard equations and a Duffing equation. Through the examples we can see the difficulty on dealing with the simple zero and the complexity in calculating of the stochastic system using the Melnikov method. So we introduce the rate of phase space flux theory to translate the problem of solving the simple zero of the Melnikov function to solving the zero of the phase space flux function, avoid the problem of solving the simple zero of the Melnikov function and predigest the calculation.更多还原