  The connection between discrete and continuous dynamical systems through the
unit-time map has shown a significant role in bifurcation theory. Recently, it
has also been used in fractal analysis of bifurcations. We study fractal
properties of the unit-time map near nilpotent nonmonodromic singularities of
planar vector fields using normal forms. We are interested in nilpotent
singularities because they are nonhyperbolic, and we know that near
nonhyperbolic singularities the box dimension is nontrivial. We study discrete
orbits generated by the unit-time map, on the separatrices at the bifurcation
point, and get results for the box dimensions of these orbits. Box dimension
results will be illustrated in details using the examples of Bogdanov-Takens
bifurcation and bifurcation of the nilpotent saddle. The study is also been
extended to the appropriate singular points at infinity of normal form for the
nilpotent singularity. Moreover, we study the unit-time map of the normal form
for a saddle, near singular points at infinity.
