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.