We study box dimension, Minkowski content and Minkowski measurability of nonrectifiable, smooth spiral trajectories of some dynamical systems in the plane. From this point of view we consider a standard model of Hopf-Takens bifurcation and study the behaviour of trajectories near singular points and limit cycles.  2005 Elsevier SAS. All rights reserved. RésuméNous etudions la « box-dimension », le contenu de Minkowski et la mesurabilité de Minkowski de la trajectoire nonrecitifiable, spirale de certains systèmes dynamiques dans le plan. De ce point de vue nous considerons une modèle standard de la bifurcation de Hopf-Takens et nous étudions le comportement des trajectoires près des points singulièrs et près des cycles limites.

Abstract. We study the asymptotics, box dimension, and Minkowski content of trajectories of some discrete dynamical systems. We show that under very general conditions, trajectories corresponding to parameters where saddle-node bifurcation appears have box dimension equal to 1/2, while those corresponding to period-doubling bifurcation parameter have box dimension equal to 2/3. Furthermore, all these trajectories are Minkowski nondegenerate. The results are illustrated in the case of logistic map.

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We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated logarithm monomials. Acknowledgement. This research was supported by: Croatian Science Foundation (HRZZ) project no. 2285, French ANR project STAAVF, French-Croatian bilateral Cogito project 33003TJ Classification de points fixes et de singularités à l'aide d'epsilon-voisinages d'orbites et de courbes, Croatian UKF project Classifications of Dulac maps and epsilon-neighborhoods 2018, the University of Zagreb research support for 2015 and 2016.