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 -