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A study of regular and chaotic motions of dynamical systems in low dimension.

机译：对低维动力系统的规则运动和混沌运动的研究。

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摘要

Chaotic systems appear in many different scientific disciplines, such as engineering, physics, chemistry, economics, biology and even political science as well as other social problems. When these systems are casually examined, some of them may repeat themselves periodically and others may present random-like chaotic behavior. The study of chaos will thus provide us with new tools to have a better understanding of such complex behavior. Since the chaotic behavior is rooted in nonlinear dynamic systems, the study of a system's nonlinearity becomes essential. Indeed, much effort has been devoted to this and the field is experiencing a rapid expansion in the past few decades.;The purpose of this dissertation is to study the behavior of nonlinear and chaotic systems. As a start, the influence of mass variation on mechanical systems will be investigated first. It will be shown that for this linear problem, a simple application of the WKB method will obtain a fairly good approximate solution. Next, a weakly nonlinear Coulomb damped system will be studied. A time varying change of variable, or the averaging method, will be employed to reduce the nonautonomous system to an autonomous one so that the response of a system with Coulomb damping can be studied in the plane described by the averaged equation.;For a highly nonlinear system, the oscillations of a pendulum with circular rotating support will be investigated. It will be shown that the system exhibits chaotic behavior. The lower bound of the chaotic regions can be determined by the Melnikov heteroclinic bifurcation criteria. The stable and unstable manifold as well as bifurcation diagram will be numerically simulated to verify the Melnikov criteria. The Liapunov spectrum will also be calculated in the forcing-frequency plane to determine the boundary of chaotic motion. It will be shown that when the system's damping is small, the Melnikov function will provide a good estimate of the heteroclinic bifurcation. For a fixed damping coefficient, the chaotic behavior will only appear for moderate values of forcing amplitude and frequency. An analysis of the chaotic region in the forcing-frequency plane will show that the effect of damping will compress the chaotic region, thus stabilize the motion. Finally, the well known Kolmogrov-Arnold-Moser (KAM) theorem will then be applied to explore the chaotic behavior of the nonintegrable Hamiltonian systems.
机译：混沌系统出现在许多不同的科学学科中，例如工程，物理，化学，经济学，生物学甚至政治学以及其他社会问题。当不经意地检查这些系统时，其中一些可能会定期重复自己，而其他系统可能会出现随机的混沌行为。因此，对混沌的研究将为我们提供新的工具，以更好地了解这种复杂的行为。由于混沌行为源于非线性动力系统，因此研究系统的非线性变得至关重要。实际上，已经为此付出了很多努力，并且在过去的几十年中，该领域正经历着快速的扩展。本论文的目的是研究非线性和混沌系统的行为。首先，将首先研究质量变化对机械系统的影响。将表明，对于该线性问题，WKB方法的简单应用将获得相当好的近似解。接下来，将研究弱非线性库仑阻尼系统。将采用随时间变化的变量或平均方法将非自治系统简化为自治系统，以便可以在由平均方程描述的平面中研究具有库仑阻尼的系统的响应。在非线性系统中，将研究具有圆形旋转支撑的摆的振动。将显示该系统表现出混沌行为。混沌区域的下界可以通过梅尔尼科夫异斜分叉标准确定。将对稳定和不稳定的歧管以及分叉图进行数值模拟，以验证梅尔尼科夫准则。 Liapunov谱也将在强迫频率平面中计算以确定混沌运动的边界。将显示出，当系统的阻尼较小时，梅尔尼科夫函数将提供对异斜分叉的良好估计。对于固定的阻尼系数，仅在强制振幅和频率为中等值时才会出现混沌行为。对受力频率平面中的混沌区域的分析将表明，阻尼效应将压缩混沌区域，从而使运动稳定。最后，众所周知的Kolmogrov-Arnold-Moser（KAM）定理将被用于探索不可积哈密顿系统的混沌行为。

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  • 作者

    Hong, Wien.;

  • 作者单位

    State University of New York at Buffalo.;

  • 授予单位 State University of New York at Buffalo.;
  • 学科 Mechanical engineering.;Mathematics.
  • 学位 Ph.D.
  • 年度 1997
  • 页码 116 p.
  • 总页数 116
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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