# A note on the box dimension of degenerate foci

## TL;DR

This paper investigates the box dimension of spiral trajectories near degenerate foci in polynomial planar systems, linking geometric complexity to system cyclicity and using transformations to analyze focus degeneracy.

## Contribution

It introduces a method to compute the box dimension of spirals near degenerate foci and explores their relation to system cyclicity under perturbations.

## Key findings

- Box dimension of spiral trajectories is computed.
- Connection established between box dimension and cyclicity.
- Transformation techniques for degenerate focus analysis are demonstrated.

## Abstract

We study polynomial planar systems with singularity of focus type without characteristic directions. Simple and natural transformation of weak focus has been used to obtain such degenerate focus. We compute the box dimension of a spiral trajectory, and show connection to cyclicity of the system under a perturbation.