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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 [1, 2), we give some sufficient conditions on the coefficients p(x) and q(x) such that every solution y=y(x),yC 2 ((0,T]) of the linear differential equation

(p(x)y ' ) ' +q(x)y=0on(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 Γ(y) of y is equal to s as well as the s dimensional upper Minkowski content (generalized length) of Γ(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(ϕ(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 '' +(μ/x)y ' +g(x)y=0,x(0,T]·

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

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34C11Qualitative theory of solutions of ODE: growth, boundedness
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