Index Theory, Stability of Orbits and Heteroclinic Phenomenon

指数理论、轨道稳定性和异宿现象

• 批准号: RGPIN-2019-06847
• 负责人: Offin, Daniel
• 金额: $ 1.24万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=716498
• 关键词: Index; Theory; Stability; Orbits; Heteroclinic

In physical systems described by a dynamical system of equations, loss of stability of an equilibrium can occur by varying the parameters in the system. For conservative systems this phenomenon is associated with parametric resonance. Instabilities are created when the frequency of an underlying stable periodic motion is in resonance with the frequency of a periodic fluctuation. The original impetus to study such systems came from the study of vibrational modes of stretched membranes Mathieu(1868) and the problem in celestial mechanics connected with determining the mean motion of the lunar perigee, Hill(1886). The early important results in the subject were already found by Mathieu(1868), Floquet(1883), Hill(1886), Rayleigh(1887) and Liapunov(1892). However these results are most commonly used to understand the stability of periodic motion which is close to some special case (integrable system) where the computations for the multipliers can be carried out. The modern direct method of variational analysis to locate periodic motion in open systems (those systems which contain a mixture of stable and chaotic motion) which are not necessarily close to an integrable case requires a new theory for determining stability or instability of such motions. In this proposal myself and my current and future graduate students are investigating new global variational and index theory methods for determining stability and instability for periodic orbits in Hamiltonian systems. From new discoveries of exo-planets outside our solar system, to current technological issues (how to stabilize a spacecraft in periodic motion around an asteroid within our solar system), the issue of detecting stable periodic orbits in high dimensional systems is an important goal of Hamiltonian dynamics. Unstable hyperbolic motion is equally important for the understanding of open systems. Indeed it is exactly this hyperbolic mechanism for periodic motion which has now become the basis for the technology of transferring spacecraft in the solar system using ballistic or gravitational attraction as an essential component of propulsion. Heteroclinic orbits describe exactly the kind of transfer in phase space between different hyperbolic or partially hyperbolic periodic orbits which is now commonly used in space flight design. My research and that of my HQP is directed towards the fundamental questions of existence and stability of periodic motion using global variational techniques. The role which symmetry may play in this topic is highly relevant since many of the recent advances in variational methods for N-body systems have used symmetry extensively. Investigation of new global techniques for finding and identifying stable and unstable motion in conservative systems, identifying new methods for detecting heteroclinic orbits and developing an understanding of the wider implications of such motions in open dynamical systems will be the focus of my research program.

在由方程的动力系统描述的物理系统中，平衡的稳定性的损失可以通过改变系统中的参数而发生。对于保守系统，这种现象与参数共振有关。当潜在的稳定周期运动的频率与周期波动的频率共振时，就会产生不稳定性。 研究这类系统的最初动力来自于Mathieu（1868）对拉伸膜振动模式的研究，以及Hill（1886）在天体力学中关于确定月球近地点平均运动的问题。 Mathieu（1868）、Floquet（1883）、Hill（1886）、Rayleigh（1887）和Liapunov（1892）已经发现了这一课题的早期重要结果。然而，这些结果是最常用的理解的稳定性的周期运动，这是接近一些特殊的情况下（可积系统），其中的乘子的计算可以进行。现代变分分析的直接方法来定位周期性运动的开放系统（这些系统包含一个混合的稳定和混乱的运动），这是不一定接近可积的情况下，需要一个新的理论来确定这种运动的稳定性或不稳定性。在这个建议中，我自己和我现在和未来的研究生正在研究新的全球变分和指数理论方法，用于确定哈密顿系统中周期轨道的稳定性和不稳定性。从太阳系外系外行星的新发现，到当前的技术问题（如何使航天器稳定地围绕太阳系内的小行星周期运动），在高维系统中检测稳定的周期轨道的问题是哈密顿动力学的一个重要目标。 不稳定双曲运动对于理解开放系统同样重要。事实上，正是这种周期运动的双曲线机制，现在已成为利用弹道或引力作为推进的基本组成部分在太阳系中转移航天器的技术的基础。异宿轨道精确地描述了不同双曲或部分双曲周期轨道之间在相空间中的转换，是目前航天设计中常用的一种方法。 我的研究和我的HQP的研究是使用全局变分技术针对周期运动的存在性和稳定性的基本问题。对称性在这一主题中可能发挥的作用是高度相关的，因为许多最近的进展在N体系统的变分方法广泛使用对称性。 调查 寻找和识别保守系统中稳定和不稳定运动的新的全球技术，确定检测异宿轨道的新方法，并发展理解 开放动力系统中这种运动的更广泛的含义将是我的研究计划的重点。