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

Box dimension of unit-time map near nilpotent singularity of planar vector field