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Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations. (English) Zbl 1201.34061

The authors consider analytic systems of differential equations in the plane whose origin is a nilpotent singularity

${x}^{\text{'}}=y+\sum _{i=1}^{\infty }{P}_{q-p+2is}\left(x,y\right),\phantom{\rule{1.em}{0ex}}{y}^{\text{'}}=\sum _{i=1}^{\infty }{Q}_{q-p+2is}\left(x,y\right),$

where $p,q$ and $n$ are natural numbers, $p\le q$, $s=\left(n+1\right)p-q>0$ and $\left({P}_{j},{Q}_{j}\right)$ are quasi-homogeneous vector fields of type $\left(p,q\right)$ and degree $j,$ with ${Q}_{\left(2n+1\right)p-q}\left(1,0\right)<0$ (necessary condition of monodromy). For them, they give an algorithm which provides the Taylor expansion of the return map near the origin. This map is computed by using special polar coordinates, already introduced by Lyapunov, associated to the solution of $dx/d\theta =-y,$ $dy/d\theta ={x}^{2n+1}$.

By using their algorithm and other tools described below, they characterize the centers and study the cyclicity of some concrete families of systems of the type given above. These tools are called by the authors conservative-dissipative decomposition of the associated vector field and the following nice result is proved: The nilpotent systems

${x}^{\text{'}}=y+{v}_{y}\left(x,y\right)K\left(v\left(x,y\right),{y}^{2}\right)+y\psi \left(v\left(x,y\right),{y}^{2}\right),\phantom{\rule{1.em}{0ex}}{y}^{\text{'}}=-{v}_{x}\left(x,y\right)K\left(v\left(x,y\right),{y}^{2}\right),$

where $v,k,\psi$ are analytic functions defined in a neighborhood of the origin and $\psi \left(0,0\right)=0,$ are integrable analytically in a neighborhood of the origin. This last result extends particular cases considered in the literature, like $v\left(x,y\right)\equiv x$ and others.

##### MSC:
 34C23 Bifurcation (ODE) 34C25 Periodic solutions of ODE 34C07 Theory of limit cycles of polynomial and analytic vector fields 37C10 Vector fields, flows, ordinary differential equations 34C05 Location of integral curves, singular points, limit cycles (ODE)