Linear estimation and distribution of the limit cycles bifurcated from a kind of degenerate polycycles
The paper studies the number and distribution of limit cycles bifurcated from a degenerate planar polycycle through three singular points: a saddle-node \(P_0\), a fine saddle \(P_1\) with finite order \(m\in\mathbb N\), and a contractive (attracting) saddle \(P_2\) with the hyperbolicity ratio \(q_2(0)\notin\mathbb Q\). The connection between \(P_0\) and \(P_1\) is of hh-type and the connection between \(P_0\) and \(P_2\) is of hp-type. It is assumed that the connection between \(P_0\) to \(P_2\) and \(P_0\) to \(P_1\) keeps unbroken. It is shown that the cyclicity of this polycycle is not more than \(3m+1\), which is linearly dependent on the order of the resonant saddle \(P_1\). It is also shown that the cyclicity is not more than \(m+3\) if \(q_2(0)>m\), and that the nearer \(q_2(0)\) is close to 1, the more limit cycles are bifurcated.
scientific article; zbMATH DE number 5164226