Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations

We derive some simple sufficient conditions on the amplitude , the phase and the instantaneous frequency such that the so-called chirp function is fractal oscillatory near a point , where and is a periodic function on . It means that oscillates near , and its graph is a fractal curve in such that its box-counting dimension equals a prescribed real number and the -dimensional upper and lower Minkowski contents of are strictly positive and finite. It numerically determines the order of concentration of oscillations of near . Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions. 1. Introduction The brilliant heuristic approach of Tricot [1] to the fractal curves such as the graph of functions and gave the main motivation for studying the fractal properties near of graph of oscillatory solutions of various types of differential equations: linear Euler-type equation (see [2]), general second-order linear equation (see [3]) where satisfies the Hartman-Wintner asymptotic condition near , half-linear equation (see [4]), linear self-adjoint equation (see [5]), and -Laplace differential equations in an annular domain (see [6]). A function is said to be a chirp function if it possesses the form , where and denote, respectively, the amplitude and phase of , and is a periodic function on . In all previously mentioned papers [2–5], authors are dealing with the fractal oscillations of second-order differential equations and are deriving some sufficient conditions on the coefficients of considered equations such that all their solutions together with the first derivative admit asymptotic behaviour near . It is formally written in the form of a chirp function, that is, and near . According to it, one can say that the asymptotic formula for solutions of considered equations satisfies the chirp-like behaviour near (on the asymptotic formula for solutions near , see [7, 8]). Then, in the dependence of a prescribed real number , authors give some asymptotic conditions on , , and such that all solutions are fractal oscillatory near with the fractal dimension . In this paper, independently of the asymptotic theory of differential equations, we firstly study the fractal oscillations of a chirp function; see Theorems 8 and 11. Second, taking two linearly independent chirp functions and , we generate some new classes of fractal oscillatory linear differential equations which are not considered in [2–5] and have the