Box dimension of a hyperbolic saddle loop

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.