Box dimension of trajectories of some discrete dynamical systems

TL;DR

This paper investigates the fractal properties of trajectories in certain discrete dynamical systems, revealing specific box dimensions associated with bifurcation types and demonstrating Minkowski nondegeneracy, with applications to the logistic map.

Contribution

It establishes general conditions under which trajectories have specific box dimensions linked to bifurcation types and confirms Minkowski nondegeneracy.

Findings

Trajectories at saddle-node bifurcation have box dimension 1/2.

Trajectories at period-doubling bifurcation have box dimension 2/3.

All studied trajectories are Minkowski nondegenerate.

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.