Applications of nonstandard analysis to dynamical systems

Homer, Barry James (1987) Applications of nonstandard analysis to dynamical systems. University of Southampton, Doctoral Thesis.

Abstract

This thesis concerns itself with some aspects of topological dynamics. We approach the subject using Non Standard Analysis as developed by Nelson. The thesis begins by developing the non standard machinery for reasoning about dynamical systems. It gives several results concerning the closedness, compactness and connectedness of shadows. The thesis then goes on to give a non standard treatment of limit sets, showing that many of the results are a simple consequence of non standard results about shadows and halos. The ideas of S-dynamical systems and s⁰-dynamical systems are then introduced. Then follows an extensive treatment of recursive concepts, giving both standard and non standard results. The areas covered include periodicity, almost periodicity, recursion, Poisson stability, minimality and their non standard characterisations. The thesis concludes by introducing the notion of a property being initial or final, a theme which runs through the thesis. The property that says the image of a halo is infintesimal or limited or unlimited is shown to be both initial and Final.

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