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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 $$\dot{x}=y(1+x^4),\;\dot{y}=-x(1+x^4)+\varepsilon P(x)y^{2m-1}$$ is studied for small $\varepsilon$. It is assumed that $P(x)$ 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 ($\varepsilon=0$) is given by $\min\{N_1, N_2, N_3\}$ where $$\cases N_1=4m+2n-2+\sin^2(n\pi/2)+[\frac14(m+n-3+\sin^2(n\pi/2))],\\ N_2=3m+2n-2-\sin^2(n\pi/2)+[\frac12(m+n-1)],\\ N_3=5m+2n-2+\sin^2(n\pi/2).\endcases$$ 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 $\Phi(h)=\oint_{x^2+y^2=h}P(x)y^{2m}dx/(1+x^4)$ which is elementary and is calculated explicitly in the paper. (Reviewer’s remark). The integral $\Phi(h)$, 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:
34C07Theory of limit cycles of polynomial and analytic vector fields
34C05Location of integral curves, singular points, limit cycles (ODE)
34C08Connections of ODE with real algebraic geometry
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References:
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