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Fractal oscillations of self-adjoint and damped linear differential equations of second-order. (English) Zbl 1244.34052

Authors’ abstract: For a prescribed real number $s\in$ [1, 2), we give some sufficient conditions on the coefficients $p\left(x\right)$ and $q\left(x\right)$ such that every solution $y=y\left(x\right),\phantom{\rule{3.33333pt}{0ex}}y\in {C}^{2}\left(\left(0,T$]) of the linear differential equation

${\left(p\left(x\right){y}^{\text{'}}\right)}^{\text{'}}+q\left(x\right)y=0\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}\left(0,T\right]$

is bounded and fractal oscillatory near $x=0$ with the fractal dimension equal to $s$. This means that $y$ oscillates near $x=0$ and the fractal (box-counting) dimension of the graph ${\Gamma }\left(y\right)$ of $y$ is equal to $s$ as well as the $s$ dimensional upper Minkowski content (generalized length) of ${\Gamma }\left(y\right)$ is finite and strictly positive. It verifies that $y$ admits similar kind of the fractal geometric asymptotic behaviour near $x=0$ like the chirp function ${y}_{ch}\left(x\right)=a\left(x\right)S\left(\varphi \left(x\right)\right)$, which often occurs in the time – frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form

${y}^{\text{'}\text{'}}+\left(\mu /x\right){y}^{\text{'}}+g\left(x\right)y=0,\phantom{\rule{3.33333pt}{0ex}}x\in \left(0,T\right]·$

In order to prove the main results, we state a new criterion for fractal oscillations near $x=0$ of real continuous functions.

MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34C11 Qualitative theory of solutions of ODE: growth, boundedness