Cyclicity of several planar graphics and ensembles through three singular points without generic conditions (Q935124)

The paper investigates the number and distribution of limit cycles bifurcating from several graphics and ensembles through a saddle-node \(P_0\) and two hyperbolic saddles \(P_1\) and \(P_2\) for the non-generic cases of \(r_1(0)=1\), \(r_2(0)\neq 1\) and \(r_1(0)\neq 1\), \(r_2(0)=1\), where \(r_1(0)\) and \(r_2(0)\) are the hyperbolicity ratio of the saddles \(P_1\) and \(P_2\), respectively. For the case of \(r_1(0)=1\), \(r_2(0)\neq 1\), it is assumed that the connection from \(P_0\) to \(P_2\) and the connection from \(P_0\) to \(P_1\) keep unbroken. It is proved that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity depends linearly on the order of the neutral saddle \(P_1\) if \(P_2\) is contractive and \(r_2(0)\in Q\). It is also shown that the nearer \(r_2(0)\) is to 1, the more limit cycles are bifurcating. For the case of \(r_1(0)\neq 1\), \(r_2(0)=1\), it is proved that these graphics and ensembles are of finite cyclicity respectively if \(P_1\) is of finite order and the hp-connection from \(P_0\) to \(P_2\) keeps unbroken. The problem is closely related to the finiteness part of Hilbert's 16th problem for quadratic fields which consists in proving the existence of a uniform bound for the number of limit cycles of quadratic plane vector fields.