11,526 research outputs found
Algebraic and analytical tools for the study of the period function
In this paper we consider analytic planar differential systems having a first integral of the form H(x, y) = A(x) + B(x)y + C(x)y2 and an integrating factor κ(x) not depending on y. Our aim is to provide tools to study the period function of the centers of this type of differential system and to this end we prove three results. Theorem A gives a characterization of isochronicity, a criterion to bound the number of critical periods and a necessary condition for the period function to be monotone. Theorem B is intended for being applied in combination with Theorem A in an algebraic setting that we shall specify. Finally, Theorem C is devoted to study the number of critical periods bifurcating from the period annulus of an isochrone perturbed linearly inside a family of centers. Four different applications are given to illustrate these results
Complexity reduction of C-algorithm
The C-Algorithm introduced in [Chouikha2007] is designed to determine
isochronous centers for Lienard-type differential systems, in the general real
analytic case. However, it has a large complexity that prevents computations,
even in the quartic polynomial case.
The main result of this paper is an efficient algorithmic implementation of
C-Algorithm, called ReCA (Reduced C-Algorithm). Moreover, an adapted version of
it is proposed in the rational case. It is called RCA (Rational C-Algorithm)
and is widely used in [BardetBoussaadaChouikhaStrelcyn2010] and
[BoussaadaChouikhaStrelcyn2010] to find many new examples of isochronous
centers for the Li\'enard type equation
The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems
In this work we study the centers of planar analytic vector fields which are
limit of linear type centers. It is proved that all the nilpotent centers are
limit of linear type centers and consequently the Poincar\'e--Liapunov method
to find linear type centers can be also used to find the nilpotent centers.
Moreover, we show that the degenerate centers which are limit of linear type
centers are also detectable with the Poincar\'e--Liapunov method.Comment: 24 pages, no figure
Isochronicity conditions for some planar polynomial systems II
We study the isochronicity of centers at for systems
where , which
can be reduced to the Li\'enard type equation. When and , using the so-called C-algorithm we found new families of
isochronous centers. When the Urabe function we provide an explicit
general formula for linearization. This paper is a direct continuation of
\cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly
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