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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.

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
Full Text: DOI
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