Fractal properties of Bessel functions

TL;DR

This paper investigates the fractal properties of solutions to second-order differential equations, specifically Bessel functions, revealing their phase dimension as 4/3 and describing their spiral behavior.

Contribution

It provides the first computation of the phase dimension for Bessel functions and their generalizations, linking fractal geometry with oscillatory differential equations.

Findings

Phase dimension of Bessel functions is 4/3.

Solutions form a wavy spiral trajectory.

Generalized Bessel equations also have computable phase dimensions.

Abstract

A fractal oscillatority of solutions of second-order differential equations near infinity is measured by oscillatory and phase dimensions. The phase dimension is defined as a box dimension of the trajectory $(x,\dot{x})$ in $\mathbb{R}^2$ of a solution $x=x(t)$, assuming that $(x,\dot{x})$ is a spiral converging to the origin. In this work, we study the phase dimension of the class of second-order nonautonomous differential equations with oscillatory solutions including the Bessel equation. We prove that the phase dimension of Bessel functions is equal to 4/3, and that the corresponding trajectory is a wavy spiral, exhibiting an interesting behavior. The phase dimension of a generalization of the Bessel equation has been also computed.