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 $ℝ$. It means that $y(x)$ oscillates near $x=x_0$, and its graph $\mathrm{\Gamma }(y)$ is a fractal curve in ${ℝ}^2$ 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.