Box dimension of a hyperbolic saddle loop

TL;DR

This paper calculates the box dimension of spiral trajectories around hyperbolic saddle loops, revealing a nuanced relationship with cyclicity and offering a potential method for analyzing complex foliations.

Contribution

It introduces a method to compute the box dimension of hyperbolic saddle loops, extending understanding beyond weak foci and limit cycles.

Findings

Box dimension relates to cyclicity but not bijectively in saddle loops.

Provides a technique for analyzing complex saddle points.

Suggests applications to foliation leaf dimensions.

Abstract

We compute the box dimension of a spiral trajectory around a hyperbolic saddle loop, as the simplest example of a hyperbolic saddle polycycle. In cases of weak foci and limit cycles, Zubrinic and Zupanovic show that the box dimension of a spiral trajectory is in a bijective correspondence with cyclicity of these sets. We show that, in saddle loop cases, the box dimension is related to the cyclicity, but the correspondence is not bijective. In addition, complex saddles are complexifications of weak foci points, as well as of hyperbolic saddles. Computing the box dimension around the saddle point of a hyperbolic saddle loop is hopefully a preliminary technique for computing the box dimension of leaves of a foliation around resonant complex saddles.