Regular and slow-fast codimension 4 saddle-node bifurcations

Using geometric singular perturbation theory, including the family blow-up as one of the main techniques, we prove that the cyclicity, i.e. maximum number of limit cycles, in both regular and slow-fast unfoldings of nilpotent saddle-node singularity of codimension 4 is 2. The blow-up technique enables us to use the well known results for slow-fast codimension 1 and 2 Hopf bifurcations, slow-fast Bogdanov-Takens bifurcations and slow-fast codimension 3 saddle and elliptic bifurcations.