Mathematical Sciences: The Dynamics of Digital Computers andMechanical Systems

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Richard McGehee proposes to carry out research on two topics. These involve the application of the theory of dynamical systems to digital circuits and a study of the dynamical systems on finite state spaces which arise from approximations of continuous systems. Richard Moeckel's work will be concerned with an investigation of collisions and of rejection to infinity in the three body problem, a study of central configurations in the n-body problem, and an investigation of normal forms near elliptic fixed points of mechanical systems. The research on finite approximation of continuous dynamical systems recognises the fact that computer experiments can necessarily only reveal approximations of the original continuous system. The focus of attention will be on questions asking whether the properties of attractors of the original system are faithfully exhibited on the video screen. McGehee's other project is to study the mathematical idealizations of the digital and analog levels of hardware and develop a theory of how digital devices can be derived from analog circuits. Moeckel's work will continue studies of periodic points and invariant curves of area preserving maps, of homoclinic phenomena and of normal forms. These investigations are all aimed towards an understanding of the possible qualitative behaviors of three point masses. The study of central configurations is important in the context of escape to infinity, multiple collisions, and the topology of energy surfaces.