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The number of limit cycles of a quintic polynomial system with center. (English) Zbl 1181.34040

The bifurcation of limit cycles in the system

$\stackrel{˙}{x}=y\left(1+{x}^{4}\right),\phantom{\rule{0.277778em}{0ex}}\stackrel{˙}{y}=-x\left(1+{x}^{4}\right)+\epsilon P\left(x\right){y}^{2m-1}$

is studied for small $\epsilon$. It is assumed that $P\left(x\right)$ is a real polynomial of degree $2n+2$ or $2n+3$ and $m,n$ are natural numbers. The main result states that an upper bound for the number of limit cycles bifurcating from the periodic orbits of the initial system ($\epsilon =0$) is given by $min\left\{{N}_{1},{N}_{2},{N}_{3}\right\}$ where

$\left\{\begin{array}{c}{N}_{1}=4m+2n-2+{sin}^{2}\left(n\pi /2\right)+\left[\frac{1}{4}\left(m+n-3+{sin}^{2}\left(n\pi /2\right)\right)\right],\hfill \\ {N}_{2}=3m+2n-2-{sin}^{2}\left(n\pi /2\right)+\left[\frac{1}{2}\left(m+n-1\right)\right],\hfill \\ {N}_{3}=5m+2n-2+{sin}^{2}\left(n\pi /2\right)·\hfill \end{array}\right\$

Moreover, there are systems with at least $3m+n-2$ limit cycles.

The proof follows from an estimation of the number of positive zeros of the integral ${\Phi }\left(h\right)={\oint }_{{x}^{2}+{y}^{2}=h}P\left(x\right){y}^{2m}dx/\left(1+{x}^{4}\right)$ which is elementary and is calculated explicitly in the paper.

(Reviewer’s remark). The integral ${\Phi }\left(h\right)$, as taken by the authors, is identically zero. One should consider a similar integral with ${y}^{2m-1}$ instead of ${y}^{2m}$ in order to obtain information about the limit cycles in the perturbed system.

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