9403774 Jones Mathematical models for the state of an evolving physical system are the subject of investigation of dynamical systems. The goal is the development of techniques for determining the eventual state of the system, whether it be of simple form, steady-state or periodic, or even chaotic. The reduction of complex systems is essential in forcing the arbitrarily complicated behavior to be susceptible to analysis. The separation of a system into fast and slow time scales affords a reduction that is particularly attractive as the complexity of the high-dimensional behavior is retained through the patching together of fast and slow structures. Such a reduction forms the context for much of Jones' proposed work. In recent work with Burns, Jones has uncovered a mechanism for the stabilization of surprising resonant motions involving the slow variation of certain parameters. This has been applied to explain the spin/orbit resonance of Mercury, and a general theory is envisaged which will be adaptable to many other problems of resonance, particularly in the context of planetary motions. In physical systems with spatial dependence, information is transported in a fixed direction by travelling waves. Jones will further develop his techniques for determining the stability of such waves, thus offering a discriminant to assess their physical observability. It is anticipated that the methods will be extended to cover the case of optical pulses, which will help in determining the feasible methods for multiple pulse transmission in nonlinear optical fibers. Nonlinear optics offers a particularly fruitful area of application as optical processes occur on a very fast time scale. The transition between different phases of materials, the sharpness of which reflects a fast spatial scale, can occur in exotic patterns. Techniques will be developed, based on the stability theory for travelling waves, for determining the underlying mechanisms for such pattern format ion.
