Wavy spirals and their fractal connection with chirps
  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. 
Luka Korkut, Domagoj Vlah, Darko Zubrinic and Vesna Zupanovic
math.CA