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Author: Cherkas, Leonid × DDC: 510 ×

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    On the construction of bifurcation curves related to limit cycles of multiplicity three for planar vector fields
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.
    For plane vector fields depending on three parameters we describe an algorithm to construct a curve in the parameter space such that to each point of this curve there belongs a vector field possessing a limit cycle of multiplicity three. One point of this curve is related to the bifurcation of a limit cycle of multiplicity three from an equilibrium point. The underlying procedure is a continuation method.
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    A new approach to study limit cycles on a cylinder
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.
    We present a new approach to study limit cycles of planar systems of autonomous differential equations with a cylindrical phase space $Z$. It is based on an extension of the Dulac function which we call Dulac-Cherkas function $Psi$. The level set $W:=vf,y) in Z: Psi(vf,y)=0$ plays a key role in this approach, its topological structure influences existence, location and number of limit cycles. We present two procedures to construct Dulac-Cherkas functions. For the general case we describe a numerical approach based on the reduction to a linear programming problem and which is implemented by means of the computer algebra system Mathematica. For the class of generalized Liénard systems we present an analytical approach associated with solving linear differential equations and algebraic equations
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    On the construction of Dulac-Cherkas functions for generalized Liénard systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.
    Dulac-Cherkas functions can be used to derive an upper bound for the number of limit cycles of planar autonomous differential systems, at the same time they provide information about their stability. In this paper we present a method to construct such functions for generalized Liénard systems by means of linear differential equations. If the degree m of the polynomial is not greater than 3, then the described algorithm works generically. By means of an example we show that this approach can be applied also to polynomials with degree m larger than 3.
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    On the approximation of the limit cycles function
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Cherkas, Leonid; Grin, Alexander; Schneider, Klaus R.
    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 procedure to approximate $l(x)$, which is based on the Newton scheme applied to the Poincaré function and represents a continuation method. Finally, we demonstrate the effectiveness of the proposed procedure by means of a Liénard system. The obtained result supports a conjecture by Lins, de Melo and Pugh.