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Study of the bifurcation of a multiple limit cycle of the second kind by means of a Dulac-Cherkas function: A case study
We consider a generalized pendulum equation depending on the scalar parameter having for = 0 a limit cycle Gamma of the second kind and of multiplicity three. We study the bifurcation behavior of Gamma for -1 ≤ ≤ (√5 + 3)/2 by means of a Dulac-Cherkas function.
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Construction of generalized pendulum equations with prescribed maximum number of limit cycles of the second kind
Consider a class of planar autonomous differential systems with cylindric phase space which represent generalized pendulum equations. We describe a method to construct such systems with prescribed maximum number of limit cycles which are not contractible to a point (limit cycles of the second kind). The underlying idea consists in employing Dulac-Cherkas functions. We also show how this approach can be used to control the bifurcation of multiple limit cycles.