The box dimension of degenerate spiral trajectories of a class of ordinary differential equations

TL;DR

This paper investigates the fractal properties, specifically the box dimension, of degenerate spiral trajectories in certain differential equations, introducing new methods and conjectures for understanding their complexity.

Contribution

It introduces the study of box dimension for degenerate spirals, formulates a conjecture for the general case, and links fractal analysis to contact point codimension.

Findings

Box dimension of polynomial degenerate focus of type (n,n) determined.

Conjecture proposed for the box dimension in the (m,n) case.

Fractal analysis of nilpotent contact points related to slow-fast spirals.

Abstract

In this paper we initiate the study of the box dimension of degenerate spiral trajectories of a class of ordinary differential equations. A class of singularities of focus type with two zero eigenvalues (nilpotent or more degenerate) has been studied. We find the box dimension of a polynomial degenerate focus of type $(n,n)$ by exploiting the well-known fractal results for $\alpha$-power spirals. In the general $(m,n)$ case, we formulate a conjecture about the box dimension of a degenerate focus. Further, we reduce the fractal analysis of planar nilpotent contact points to the study of the box dimension of a slow-fast spiral generated by their "entry-exit" function. There exists a bijective correspondence between the box dimension of the slow-fast spiral and the codimension of contact points. We also construct a three-dimensional vector field that contains a degenerate spiral, called an elliptical power spiral, as a trajectory.