Fractal analysis of spiral trajectories of some planar vector fields

Abstract

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.

Rectifiable oscillations in second-order linear differential equations

“…Furthermore, in the case of nondegeneracy of the Minkowski content, the s-sets defined in Definition 1.11 appear also as Minkowski measurability, which is an important concept in the study of measurable fractals and eigenfrequencies of fractal strings and fractal drums (Weyl-Berry conjecture), see [3,8], and [9]. Minkowski measurability has been studied in the context of dynamical systems, see [21].…”

“…We know that the solutionẑ(s) of Eq. (20) satisfies the asymptotic behaviour(21). Now, let us choose x k such that…”

Box-counting dimensions of generalised fractal nests

“…The well understood type of behaviour of fractal nests concerns sets that locally resemble Cartesian products of fractals [LRŽ17,pg. 227], [ŽŽ05,Remark 6.]. In such sets, the dimension behaves naturally as the sum of the dimensions of "base" sets.…”

Fractal properties of Bessel functions

“…The standard Bessel equation now becomes a specific case of this generalization. In order to prove Theorem 1, we first obtain a new version of some theorems from [13].…”