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Zbl 1244.34052
Pašić, Mervan; Tanaka, Satoshi
Fractal oscillations of self-adjoint and damped linear differential equations of second-order.
(English)
[J] Appl. Math. Comput. 218, No. 5, 2281-2293 (2011). ISSN 0096-3003

Authors' abstract: For a prescribed real number $s \in$ [1, 2), we give some sufficient conditions on the coefficients $p(x)$ and $q(x)$ such that every solution $y = y(x), ~y \in C^{2}((0, T$]) of the linear differential equation $$(p(x)y^{\prime})^{\prime} + q(x)y = 0 \text{ on }(0, T]$$ 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 $\Gamma (y)$ of $y$ is equal to $s$ as well as the $s$ dimensional upper Minkowski content (generalized length) of $\Gamma (y)$ 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 $y_{ch}(x) = a(x)S(\varphi (x))$, 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 $$y''+ (\mu /x)y^{\prime} + g(x)y = 0,~ x \in (0, T].$$ In order to prove the main results, we state a new criterion for fractal oscillations near $x = 0$ of real continuous functions.
[Qingkai Kong (DeKalb)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: linear equations; asymptotic behaviour of solutions; oscillations; chirps; fractal curves; fractal dimension; Minkowski content; Bessel equation

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