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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… (More)

- Alexander Grin, Klaus R. Schneider
- I. J. Bifurcation and Chaos
- 2016

We consider a generalized pendulum equation depending on the scalar parameter μ having for μ = 0 a limit cycle Γ of the second kind and of multiplicity three. We study the bifurcation behavior of Γ for −1 ≤ μ ≤ ( √ 5 + 3)/2 by means of a Dulac-Cherkas function.

We consider planar vector fields depending on a real parameter. It is assumed that this vector field has a family of limit cycles which can be described by means of the limit cycles function l. We prove a relationship between the multiplicity of a limit cycle of this family and the order of a zero of the limit cycles function. Moreover, we present a… (More)

- Leonid Cherkas, Alexander Grin, Klaus R. Schneider
- J. Computational Applied Mathematics
- 2013

- Alexander Grin, Klaus R. Schneider
- I. J. Bifurcation and Chaos
- 2012

Consider a polynomial Liénard system depending on three parameters a, b, c and with the following properties: (i) The origin is the unique equilibrium for all parameters. (ii). If a crosses zero, then the origin changes its stability, and a limit cycle bifurcates from the equilibrium. We investigate analytically this bifurcation in dependence on the… (More)

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