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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Rotation intervals for chaotic sets
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by Kathleen T. Alligood and James A. Yorke PDF
Proc. Amer. Math. Soc. 126 (1998), 2805-2810 Request permission

Abstract:

Chaotic invariant sets for planar maps typically contain periodic orbits whose stable and unstable manifolds cross in grid-like fashion. Consider the rotation of orbits around a central fixed point. The intersections of the invariant manifolds of two periodic points with distinct rotation numbers can imply complicated rotational behavior. We show, in particular, that when the unstable manifold of one of these periodic points crosses the stable manifold of the other, and, similarly, the unstable manifold of the second crosses the stable manifold of the first, so that the segments of these invariant manifolds form a topological rectangle, then all rotation numbers between those of the two given orbits are represented. The result follows from a horseshoe-like construction.
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Additional Information
  • Kathleen T. Alligood
  • Affiliation: Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030
  • Email: alligood@gmu.edu
  • James A. Yorke
  • Affiliation: Institute for Physical Sciences and Technology, University of Maryland, College Park, Maryland 20742
  • Email: yorke@ipst.umd.edu
  • Received by editor(s): January 24, 1997
  • Additional Notes: The authors’ research was partially supported by the National Science Foundation. The second author’s research was also supported by the Department of Energy (Office of Scientific Computing)
  • Communicated by: Linda Keen
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 2805-2810
  • MSC (1991): Primary 58Fxx
  • DOI: https://doi.org/10.1090/S0002-9939-98-04267-1
  • MathSciNet review: 1443368