Fractal oscillations of self-adjoint and damped linear differential equations of second-order

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), y ∈ C2((0, T]) of the linear differential equation (p(x)y′)′ + q(x)y = 0 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 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 ych(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 which essentially improves related one presented in [1].