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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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Bifurcation of limit cycles: geometric theory
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by L. M. Perko PDF
Proc. Amer. Math. Soc. 114 (1992), 225-236 Request permission

Abstract:

Multiple limit cycles play a basic role in the theory of bifurcations. In this paper we distinguish between singular and nonsingular, multiple limit cycles of a system defined by a one-parameter family of planar vector fields. It is shown that the only possible bifurcation at a nonsingular, multiple limit cycle is a saddle-node bifurcation and that locally the resulting stable and unstable limit cycles expand and contract monotonically as the parameter varies in a certain sense. Furthermore, this same type of geometrical behavior occurs in any one-parameter family of limit cycles experiencing a saddle-node type bifurcation except possibly at a finite number of points on the multiple limit cycle.
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Additional Information
  • © Copyright 1992 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 114 (1992), 225-236
  • MSC: Primary 34C23; Secondary 34C05, 34C25
  • DOI: https://doi.org/10.1090/S0002-9939-1992-1086341-1
  • MathSciNet review: 1086341