WIAS Preprint No. 2272, (2016)

Construction of generalized pendulum equations with prescribed maximum number of limit cycles of the second kind


  • Schneider, Klaus R.
  • Grin, Alexander

2010 Mathematics Subject Classification

  • 34C05 34C23


  • generalized pendulum equation, limit cycles of the second kind, limit cycle of multiplicity three, bifurcation behavior, Dulac--Cherkas function




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.

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