Fractal oscillations of chirp functions and applications to second-order linear differential equations
type: Item
instance of: scholarly article
title: Fractal oscillations of chirp functions and applications to second-order linear differential equations
zbMATH Open document ID: 1269.34040
DOI: 10.1155/2013/857410
author: Mervan Pašić
author: Satoshi Tanaka
published in: International Journal of Differential Equations
publication date: 2013-06-10T00:00:00Z
review text: Summary: We derive some simple sufficient conditions on the amplitude \(a(x)\), the phase \(\varphi(x)\) and the instantaneous frequency \(\omega(x)\) such that the so-called chirp function \(y(x)=a(x)S(\varphi(x))\) is fractal oscillatory near a point \(x = x_0\), where \(\varphi'(x)=\omega(x)\) and \(S=S(t)\) is a periodic function on \(\mathbb R\). It means that \(y(x)\) oscillates near \(x=x_0\), and its graph \(\Gamma(y)\) is a fractal curve in \(\mathbb R^2\) such that its box-counting dimension equals a prescribed real number \(s \in [1, 2)\) and the \(s\)-dimensional upper and lower Minkowski contents of \(\Gamma(y)\) are strictly positive and finite. It numerically determines the order of concentration of oscillations of \(y(x)\) near \(x = x_0\). Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions.
Mathematics Subject Classification ID: 34C10
Mathematics Subject Classification ID: 34A08
Mathematics Subject Classification ID: 34A30
zbMATH DE Number: 6172138
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2013/857410
OpenAlex ID: W2010061987
Wikidata QID: Q58923642
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description: scientific article; zbMATH DE number 6172138