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On a class of partial differntial equations with hysteresis arising in magnetohydrodynamics
(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2007) Eleuteri, Michela
In this paper we deal with a class of parabolic partial differential equations containing a continuous hysteresis operator. We get an existence result by means of a technique based on an implicit time discretization scheme and we also analyse the dependence of the solution on the data. This model equation appears in the context of magnetohydrodynamics.
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.