First-order perturbation for multi-parameter center families
It is a famous problem to find the exact upper bound \(Z(n)\) of the number of zeroes of the integral \[M(h)=\int_{\{H(x, y)=h\}}\frac{Q(x,y)dy-P(x,y)dx}{V(x,y)},\] where \(H, V, P, Q\) are polynomials of degree \(n\) and the curve \(\{H(x, y)=h\}\) are ovals. In this paper the authors try to give a lower bound of \(Z(n)\). Their idea is: add a small parameter \(\lambda\), so that \(M(h)\) can be writen as \[M(h)=M_0(h)+\lambda M_1(h)+\lambda^2 M_2(h)+\ldots\] Obviusly, if \(M_0(h)=M_1(h)=\ldots=M_{k-1}(h)=0\), but \(M_k(h)\ne 0\), then the number of \(M_k(h)\) can give a lower bound of the number of zeros of \(M(h)\). At last, the authors show that \(M_k(h)\) may have more zeros than \(M_0(h)\). A very similar idea has been appeared in the following paper [\textit{M. Han} and \textit{Y. Xiong}, Chaos Solitons Fractals 68, 20--29 (2014; Zbl 1354.34062)], but the authors do not notice this paper.
scientific article; zbMATH DE number 7450439