Fractal oscillations of chirp functions and applications to second-order linear differential equations (CROSBI ID 191272)
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Podaci o odgovornosti
Pašić, Mervan ; Tanaka, Satoshi
engleski
Fractal oscillations of chirp functions and applications to second-order linear differential equations
We derive some simple sufficient conditions on the amplitude $a(x)$, the phase $\varphi (x)$ and the instantaneous frequency $\omega (x)$ such that the so-called chirp function $y(x)=a (x)\, S(\varphi (x))$ is fractal oscillatory near a point $x=x_0$, where $\varphi' (x)=\omega(x)$ and $S=S(t)$ is a periodic function on $\mathbb{;R};$. It means that $y(x)$ oscillates near $x=x_0$ and its graph $\Gamma (y)$ is a fractal curve in $\mathbb{;R};^2$ such that its box-counting dimension equals to a prescribed real number $s\in [1, 2)$ and the $s$-dimensional upper and lower Minkowski contents of $\Gamma (y)$ 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.
chirp; fractal; oscillation; box dimension; differential equations
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Podaci o izdanju
2013
2013.
1-11
objavljeno
1687-9643
1687-9651
http://www.hindawi.com/journals/ijde/2013/857410/
Povezanost rada
Temeljne tehničke znanosti, Matematika