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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.