Wavy spirals and their fractal connection with chirps

Abstract

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 $r=f(\phi )$, converging to the origin in non-monotone way as a function of $\phi$. They emerged in our study of phase portraits associated to solutions of Bessel equations. Also, the rectifiable chirps and spirals have been studied.

References in corpus (1)

• Geometrical properties of a class of systems with spiral trajectories in R^3

Cited by in corpus (1)

• Geometrical properties of a class of systems with spiral trajectories in R^3