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

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), y ∈ C 2((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 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 which essentially improves related one presented in [1].