engleski

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 $4/3$, for each order of the Bessel function. A trajectory is a wavy spiral, exhibiting an interesting oscillatory behavior. The phase dimension of a generalization of the Bessel equation has been also computed.

wavy spiral ; Bessel equation ; generalized Bessel equation ; box dimension ; phase dimension

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