The number of limit cycles of a quintic polynomial system with center
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 \[ \begin{cases} N_1=4m+2n-2+\sin^2(n\pi/2)+[\frac14(m+n-3+\sin^2(n\pi/2))],\\ N_2=3m+2n-2-\sin^2(n\pi/2)+[\frac12(m+n-1)],\\ N_3=5m+2n-2+\sin^2(n\pi/2).\end{cases} \] 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.
scientific article; zbMATH DE number 5586835