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Fractal analysis of unit time map and cyclicity of nilpotent singularities of planar vector field

This article shows how fractal analysis of the unit-time map can be used in studying the cyclicity problem of nilpotent singularities. We study fractal properties such as box dimension and "-neighbourhood of discrete orbits generated by the unit-time map. In the case of bifurcations of non- hyperbolic singularities such as saddle-node or Hopf-Takens bifurcation, there is already known connection between the multiplicity of singularity and the box dimension of the unit-time map or Poincar´e map. In this article we study how the box dimension and "- neighbourhood of discrete orbits generated by the unit-time map near nilpotent singularities are connected to the known bounds for local cyclicity of singularities. In this analysis, we use the restriction of the unit-time map on the characteristic curves or separatrices, depending on the type of singularity. Main nilpotent singularities which are studied here are nilpotent node, focus and cusp. Moreover, we study fractal properties of the unit-time map for nilpotent singularities at infinity.

box dimension; nilpotent singularity

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