Bifurcation of limit cycles: geometric theory

Perko, L. M.

doi:10.1090/S0002-9939-1992-1086341-1

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.

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