Finite cyclicity of loops and cusps

The Hilbert 16th problem for polynomial planar vector fields is a consequence of the following conjecture: any analytic deformation of the planar vector field germ along a limit periodic set has a finite cyclicity (a finite bound for the number of limit cycles near the limit periodic set). In this paper the property of finite cyclicity is established for the simplest singular limit sets: loops made by homoclinic connection at a hyperbolic saddle point and cuspidal singular points.