## International Journal of Differential Equations

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

#### Abstract

We derive some simple sufficient conditions on the amplitude $a(x)$, the phase $\phi (x),$ and the instantaneous frequency $\omega (x)$ such that the so-called chirp function $y(x)=a(x)S(\phi (x))$ is fractal oscillatory near a point $x={x}_{0}$, where ${\phi }^{\prime }(x)=\omega (x)$ and $S=S(t)$ is a periodic function on $\Bbb R$. It means that $y(x)$ oscillates near $x={x}_{0}$, and its graph $\mathrm{\Gamma }(y)$ is a fractal curve in ${\Bbb R}^{2}$ such that its box-counting dimension equals a prescribed real number $s\in [1,2)$ and the $s$-dimensional upper and lower Minkowski contents of $\mathrm{\Gamma }(y)$ 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.

#### Article information

Source
Int. J. Differ. Equ., Volume 2013, Special Issue (2013), Article ID 857410, 11 pages.

Dates
Accepted: 8 January 2013
First available in Project Euclid: 24 January 2017

https://projecteuclid.org/euclid.ijde/1485226924

Digital Object Identifier
doi:10.1155/2013/857410

Mathematical Reviews number (MathSciNet)
MR3038080

Zentralblatt MATH identifier
1269.34040

#### Citation

Pašić, Mervan; Tanaka, Satoshi. Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations. Int. J. Differ. Equ. 2013, Special Issue (2013), Article ID 857410, 11 pages. doi:10.1155/2013/857410. https://projecteuclid.org/euclid.ijde/1485226924

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