The generalized center problem of degenerate resonant singular point
The paper is concerned with generalized center condition and integrability of the origin for complex polynomial differential equation system \[ \left\{ \begin{aligned} \frac{dz}{dT}&=pz^{n+1}w^n+ \sum_{\alpha+\beta=2n+2}^{\infty}a_{\alpha\beta}z^{\alpha}w^{\beta},\\ \frac{dw}{dT}&=-qw^{n+1}z^n- \sum_{\alpha+\beta=2n+2}^{\infty}b_{\alpha\beta}z^{\alpha}w^{\beta},\\ \end{aligned} \right. \eqno (1) \] where \(z\), \(w\), \(T\) are complex, \(a_{\alpha\beta}\), \(b_{\alpha\beta}\) are comlex constants, \(p,q\in\mathbb Z^{+}\), \((p,q)=1\), \(n\) is a positive integer. The origin is a degenerate \(p:-q\) resonant singular point of system (1). The method based on a homeomorphic transformation of the degenerate singular point into the elementary one allows to compute the generalized singular point quantities and determine the generalized center condition for the origin. Necessary and sufficient conditions of generalized complex center of degenerate resonant singular point are obtained.
scientific article; zbMATH DE number 5527927