  We study the fractal oscillatority of a class of real $C^1$ functions
$x=x(t)$ near $t=\infty$. It is measured by oscillatory and phase dimensions,
defined as box dimensions of the graph of $X(\tau)=x(\frac{1}{\tau})$ near
$\tau=0$ and the trajectory $(x,\dot{x})$ in $\mathbb{R}^2$, respectively,
assuming that $(x,\dot{x})$ is a spiral converging to the origin. The
relationship between these two dimensions has been established for a class of
oscillatory functions using formulas for box dimensions of graphs of chirps and
nonrectifiable wavy spirals, introduced in this paper. Wavy spirals are a
specific type of spirals, given in polar coordinates by $r=f(\varphi)$,
converging to the origin in non-monotone way as a function of $\varphi$. They
emerged in our study of phase portraits associated to solutions of Bessel
equations. Also, the rectifiable chirps and spirals have been studied.
