Vol. 51, No. 1, 1974

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Asymptotic approach to periodic orbits and local prolongations of maps

Robert John Sacker

Vol. 51 (1974), No. 1, 273–287
Abstract

This paper is concerned with a flow on a metric space, and some topological properties of the set of orbits which are asymptotic to a given invariant subset, with particular emphasis on the flow near an invariant Jordan curve (e.g., a periodic orbit) in an orientable n-manifold Mn. The invesligation began with the asking of the simple question: Can a periodic orbit J of a vector field in Rn be the ω-limit set of precisely one orbit distinct from J? It is shown that if the periodic orbit J is a maximal element in the class of invariant continuua lying in a neighborhood of J, then the answer is negative and in fact the set of orbits asymptotic to J as t →∞ has some of the same topological properties already known from the stable manifold theorems in the case of an elementary periodic orbit of a flow generated by a smooth ordinary differential equation.

Mathematical Subject Classification 2000
Primary: 58F20
Secondary: 54H20
Milestones
Received: 30 November 1972
Published: 1 March 1974
Authors
Robert John Sacker