## Bifurcation of limit cycles: geometric theory

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- by L. M. Perko
- Proc. Amer. Math. Soc.
**114**(1992), 225-236 - DOI: https://doi.org/10.1090/S0002-9939-1992-1086341-1
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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.## References

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## Bibliographic 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