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

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.