Index Theory, Stability of Orbits and Heteroclinic Phenomenon

在由动力学方程组描述的物理系统中，改变系统中的参数可能导致平衡稳定性的丧失。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。研究这种系统的最初动力来自于对拉伸膜振动模式的研究，Mathieu（1868）和天体力学中与确定月球近地点平均运动有关的问题Hill（1886）。该学科早期的重要成果已经由Mathieu（1868）、Floquet（1883）、Hill（1886）、Rayleigh（1887）和Liapunov（1892）发现。然而，这些结果最常用于理解周期运动的稳定性，它接近于一些特殊情况（可积系统），在这些情况下可以进行乘子的计算。现代直接变分分析方法定位开放系统（那些包含稳定和混沌运动的混合系统）中的周期运动，这些运动不一定接近于可积情况，需要一种新的理论来确定这种运动的稳定性或不稳定性。在这个提案中，我和我现在和未来的研究生正在研究新的全局变分和指标理论方法来确定哈密顿系统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（如何稳定太阳系内小行星周围周期性运动的航天器），探测高维系统中稳定的周期性轨道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机制现在已经成为利用弹道或重力吸引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。异斜轨道准确地描述了不同双曲或部分双曲周期轨道之间的相空间转移，这种转移目前在航天飞行设计中常用。我的研究和我的HQP的研究是针对使用全局变分技术的周期运动的存在和稳定性的基本问题。对称性在这个主题中可能扮演的角色是高度相关的，因为最近在n体系统变分方法中的许多进展都广泛使用了对称性。研究发现和识别保守系统中稳定和不稳定运动的新全球技术，确定检测异斜轨道的新方法，并发展对开放动力系统中此类运动的更广泛含义的理解，将是我研究计划的重点。

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.