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arXiv · 1210.6611

Wavy spirals and their fractal connection with chirps

Luka Korkut·Domagoj Vlah·Darko Zubrinic·Vesna Zupanovic
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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 $X(τ)=x(\frac{1}τ)$ near $τ=0$ 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 $r=f(φ)$, converging to the origin in non-monotone way as a function of $φ$. They emerged in our study of phase portraits associated to solutions of Bessel equations. Also, the rectifiable chirps and spirals have been studied.

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Luka Korkut, Domagoj Vlah, Darko Zubrinic, Vesna Zupanovic. 2014-04-21. Wavy spirals and their fractal connection with chirps. https://arxiv.org/abs/1210.6611

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