In this thesis we study perturbations from planar vector fields having a line of zeros and representing a singular limit of Bogdanov– Takens (BT) bifurcations . We introduce , among other precise definitions , the notion of slow–fast BT-bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits . Based on geometric singular perturbation theory , including blow-up , we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane. We study the cyclicity of limit periodic sets that occur in families of vector fields of slow–fast type . The limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point . At this turning point a stability change takes place : on one side of the turning point the dynamics point strongly towards the curve of singularities; on the other side the dynamics point away from the curve of singularities . The presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point , i.e . orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve . All existing results deal with a non-zero slow movement , permitting a good estimate of the cyclicity by considering the slow-divergence integral along the curve of singularities. The most difficult problem to deal with concerns the uniform treatment of the evolution that a limit cycle undergoes when it grows from a small limit cycle near the singular point to a canard cycle of detectable size; i.e . a limit cycle that its hausdorff limit is not a single point but a canard limit periodic set (or a slow-fast cycle).