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Fractal Analysis of unit-time map and cyclicity of nilpotent singularities of planar vector fields

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 epsilon-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 Poincare map. In this article we study how the box dimension and epsilon-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.

unit-time map; box dimension; nilpotent singularity; cyclicity; node; focus

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