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Bifurcation of limit cycles from a polynomial non-global center. (English) Zbl 1166.34017

The paper studies the number of limit cycles that bifurcate, when $\epsilon$ is small enough, from the period annulus of the system

$\stackrel{˙}{x}=-yF\left(x,y\right)+\epsilon P\left(x,y\right),\phantom{\rule{1.em}{0ex}}\stackrel{˙}{y}=xF\left(x,y\right)+\epsilon Q\left(x,y\right),$

$P\left(x,y\right)$ and $Q\left(x,y\right)$ are arbitrary real polynomials of degree $n$. The main subject is the situation when $F\left(x,y\right)=0$ is formed by $k$ non-zero singular points.

The main goal is to give lower and upper bounds for the zeros for corresponding Abelian integral in terms of $k$ and $n$. One of the key points is that the Abelian integral can be explicitly obtained as an application of the integral representation formula of harmonic functions through the Poisson kernel.

##### MSC:
 34C07 Theory of limit cycles of polynomial and analytic vector fields 34C23 Bifurcation (ODE) 34C05 Location of integral curves, singular points, limit cycles (ODE)
##### References:
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