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Limit cycles appearing from the perturbation of a system with a multiple line of critical points. (English) Zbl 1241.34037

Consider the family of planar systems

$\stackrel{˙}{x}=-yC\left(x,y\right)+ϵP\left(x,y\right);\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\stackrel{˙}{y}=xC\left(x,y\right)+ϵQ\left(x,y\right)$

where $P,Q$ and $C$ are real polynomials, $C\left(0,0\right)\ne 0$, and $ϵ$ is a small real parameter. The number of zeros of the abelian integral $M\left(r\right)$ on ${x}^{2}+{y}^{2}={r}^{2}$ controls the number of limit cycles that bifurcate from the periodic orbits of the unperturbed system with $ϵ=0$. The authors consider the case $C\left(x,y\right)={\left(1-y\right)}^{m}$ and $P,Q$ are of degree $n$. They prove that $M\left(r\right)=\left[\frac{m+n}{2}\right]-1$ when $n and $n$ when $n\ge m-1$.

##### MSC:
 34C08 Connections of ODE with real algebraic geometry 34C07 Theory of limit cycles of polynomial and analytic vector fields 34C23 Bifurcation (ODE) 34C05 Location of integral curves, singular points, limit cycles (ODE)