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Fractal oscillations of chirp functions and second-order differential equations

A function $y(x)=a (x)\, S(\varphi (x))$ is called a chirp function, where $a(x)$ and $\varphi (x)$ denote respectively the amplitude and phase of $y(x)$ and $S=S(t)$ is a periodic function on $\mathbb{;R};$. For an arbitrary real number $s\in [1, 2)$, we find some simple asymptotic conditions on $a(x)$ and $\varphi (x)$ near $x=0$ such that the chirp function $y(x)$ is fractal oscillatory near $x=0$. It means that $y(x)$ oscillates near $x=0$ and its graph $\Gamma (y)$ is a fractal curve in $\mathbb{;R};^2$, that is, the box-counting dimension of $\Gamma (y)$ is equal to $s$ and the $s$-dimensional upper Minkowski content of $\Gamma (y)$ is strictly positive and finite. The fractal oscillations have been recently introduced in the case of the graph of oscillatory solutions of several types of differential equations: linear Euler type equation $y''+\lambda x^{;-\sigma};y=0$ (see Pa\v{;s};i\'c in 2008), general second-order linear equation $y''+f(x)y=0$ (see Kwong, Pa\v{;s};i\'c and Wong in 2008), where $f(x)$ satisfies the Hartman-Wintner asymptotic condition near $x=0$ (see the books by Coppel from 1965 and Hartman from 1982), half-linear equation $(|y'|^{;p-2};y')'+f(x)|y|^{;p-2};y=0$ (see Pa\v{;s};i\'c and Wong in 2009), and linear self-adjoint equation $(p(x)y')'+q(x)y=0$ (see Pa\v{;s};i\'c and Tanaka in 2011).

chirp; fractal oscillations; linear differential equations

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