TY - JOUR

T1 - Fractal oscillations of chirp functions and applications to second-order linear differential equations

AU - Pašić, Mervan

AU - Tanaka, Satoshi

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013

Y1 - 2013

N2 - We derive some simple sufficient conditions on the amplitude a (x), the phase φ (x), and the instantaneous frequency ω (x) such that the so-called chirp function y (x) = a (x) S (φ (x)) is fractal oscillatory near a point x = x 0, where φ ′ (x) = ω (x) and S = S (t) is a periodic function on. It means that y (x) oscillates near x = x 0, and its graph Γ (y) is a fractal curve in R2 such that its box-counting dimension equals a prescribed real number s ε [ 1, 2) and the s -dimensional upper and lower Minkowski contents of Γ (y) are strictly positive and finite. It numerically determines the order of concentration of oscillations of y (x) near x = x 0. 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.

AB - We derive some simple sufficient conditions on the amplitude a (x), the phase φ (x), and the instantaneous frequency ω (x) such that the so-called chirp function y (x) = a (x) S (φ (x)) is fractal oscillatory near a point x = x 0, where φ ′ (x) = ω (x) and S = S (t) is a periodic function on. It means that y (x) oscillates near x = x 0, and its graph Γ (y) is a fractal curve in R2 such that its box-counting dimension equals a prescribed real number s ε [ 1, 2) and the s -dimensional upper and lower Minkowski contents of Γ (y) are strictly positive and finite. It numerically determines the order of concentration of oscillations of y (x) near x = x 0. 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.

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U2 - 10.1155/2013/857410

DO - 10.1155/2013/857410

M3 - Article

AN - SCOPUS:84887812349

VL - 2013

JO - International Journal of Differential Equations

JF - International Journal of Differential Equations

SN - 1687-9643

M1 - 857410

ER -