Fractal properties of Bessel functions

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.