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Poincare map in fractal analysis of spiral trajectories of planar vector fields

We study the box dimension and Minkowski content of spiral trajectories of planar vector fields, using information about the asymptotic behaviour of iterates of the Poincar\'e map. An auxilliary tool is a flow-sector theorem near the weak focus, which is of a similar nature as the well known flow-box theorem. Applications include Hopf bifurcation, Li\'enard systems, and weakly damped oscillators. We obtain all possible values of box dimensions of spiral trajectories around weak focus, associated with polynomial vector fields.

Poincar\'e map; spiral; box dimension; flow-sector; Hopf bifurcation; Li\'enard system; weakly damped oscillator

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