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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Bifurcation of limit cycles: geometric theory


Author: L. M. Perko
Journal: Proc. Amer. Math. Soc. 114 (1992), 225-236
MSC: Primary 34C23; Secondary 34C05, 34C25
MathSciNet review: 1086341
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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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DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1086341-1
PII: S 0002-9939(1992)1086341-1
Article copyright: © Copyright 1992 American Mathematical Society