Index Theory, Stability of Orbits and Heteroclinic Phenomenon

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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.