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

This paper investigates the number and distribution of the limit cycles bifurcated from several graphics and ensembles through a saddle-node P0 and two hyperbolic saddles P1 and P2 for the non-generic cases of r1(0)=1, r2(0)≠1 and r1(0)≠1, r2(0)=1, where r1(0) and r2(0) are the hyperbolicity ratio of the saddles P1 and P2, respectively. For the case of r1(0)=1, r2(0)≠1, we suppose that the connection from P0 to P2 and the connection from P0 to P1 keep unbroken. We prove that these graphics and ensembles are of finite cyclicity respectively. Moreover, the cyclicity is linearly dependent on the order of the neutral saddle P1 if P2 is contractive and r2(0)∈Q. We also show that the nearer r2(0) is close to 1, the more the limit cycles are bifurcated. For the case of r1(0)≠1, r2(0)=1, we obtain that these graphics and ensembles are of finite cyclicity respectively if P1 is of finite order and the hp-connection from P0 to P2 keeps unbroken.