Abstract
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.
Highlights
The brilliant heuristic approach of Tricot [1] to the fractal curves such as the graph of functions y(x) = xα sin x−β and y(x) = xα cos x−β gave the main motivation for studying the fractal properties near x = 0 of graph of oscillatory solutions of various types of differential equations: linear Euler-type equation y + λx−σy = 0, general second-order linear equation y + f(x)y = 0 where f(x) satisfies the Hartman-Wintner asymptotic condition near x = 0, halflinear equation (|y|p−2y) + f(x)|y|p−2y = 0, linear self-adjoint equation (p(x)y) + q(x)y = 0, and pLaplace differential equations in an annular domain
In all previously mentioned papers [2,3,4,5], 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 y(x) together with the first derivative y(x) admit asymptotic behaviour near x = 0
One can say that the asymptotic formula for solutions of considered equations satisfies the chirp-like behaviour near x = 0
Summary
In all previously mentioned papers [2,3,4,5], 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 y(x) together with the first derivative y(x) admit asymptotic behaviour near x = 0. It is formally written in the form of a chirp function, that is, y(x) = a(x) sin(φ(x)) and y(x) = b(x) cos(φ(x)) near x = 0. The chirp functions are appearing in the timefrequency analysis; see for instance, [11,12,13,14,15] as well as in several applications of the time-frequency analysis; see for instance, [16,17,18,19,20]
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