Cyclicity of a kind of degenerate polycycles through three singular points (Q2496382)

This paper deals with the cyclicity of a kind of degenerate planar polycycles through a saddle-node \(P_{0}\) and two hyperbolic saddles \(P_{1}\) and \(P_{2},\) where the hyperbolicity ratio of the saddle \(P_{1}\) is equal to~1 and that of the other saddle \(P_{2}\) is irrational. It is assumed that the connections between \(P_{0}\) to \(P_{1}\) and \(P_{0}\) to \(P_{2}\) keep unbroken. The author proves that the cyclicity of this kind of polycycles is not larger than \(m+3\) if the saddle \(P_{1}\) is of order~\(m\) and the hyperbolicity ratio of \(P_{2}\) is bigger than~\(m.\) Furthermore, the cyclicity of this polycycle is not larger than~7 if the saddle \(P_{1}\) is of order~2 and the hyperbolicity ratio of \(P_{2}\) is located in the interval \((1,2).\)