Fractal analysis of Hopf bifurcation at infinity

TL;DR

This paper extends the concept of box dimension to unbounded sets using geometric inversion and stereographic projection, and applies this to analyze the Hopf-Takens bifurcation at infinity.

Contribution

It introduces a new approach to measure fractal dimensions of unbounded sets and applies it to bifurcation analysis at infinity.

Findings

Extended box dimension definition for unbounded sets

Applied fractal analysis to Hopf-Takens bifurcation at infinity

Provided properties and equivalences of the new dimension measure

Abstract

Using geometric inversion with respect to the origin we extend the definition of box dimension to the case of unbounded subsets of Euclidean spaces. Alternative but equivalent definition is provided using stereographic projection on the Riemann sphere. We study its basic properties, and apply it to the study of the Hopf-Takens bifurcation at infinity.