Vol. 51, No. 1, 1974

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Asymptotic approach to periodic orbits and local prolongations of maps

Robert John Sacker

Vol. 51 (1974), No. 1, 273–287

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
Received: 30 November 1972
Published: 1 March 1974
Robert John Sacker