engleski

Fractal oscillations: details, closed and open problems

In the first part, the definitions and some explicit examples of real functions and linear differential equations which oscillate near $x=0$ are presented. Also, some very simple sufficient and necessary conditions are given on the coefficient $f(x)$ of the equation $(P)$: $y''+f(x)y=0$, $x\in (0, 1]$, such that the equation $(P)$ is oscillatory near $x=0$. As a consequence, it is shown that many well-known classes of the linear differential equations are oscillating near $x=0$ like Euler equation, Riemann-Weber version of Euler equation and the $(\alpha, \beta)$-chirp equation. Next, we discuss the notion of 'a chirp' which appears in the nature as well as in the acoustic and signal processing, and in the mathematics. We discuss why the chirps give the main motivation to introduce and to study the fractal oscillations of real functions and differential equations on the unit interval $[0, 1]$. Besides the length of graph $G(y)$ of a chirp $y(x)$, the fractal (box-counting) dimension $\dim_{; ; M}; ; G(y)$ and corresponding Minkowski content $M^{; ; s}; ; (G(y))$ of $G(y)$ play essential role to estimate the density of an area filled by an oscillating chirp near $x=0$. Next, we present the first main result on the fractal oscillations obtained for the Euler type equation $y''+\lambda x^{; ; -\sigma}; ; y=0$, $x\in (0, 1]$, $\lambda >0$, $\sigma\geq 2$ (see M. Pa\v{; ; s}; ; i\'{; ; c}; ; , 'Fractal oscillations for a class of second-order linear differential equations of Euler type', J. Math. Anal. Appl. 341 (2008) 211-223.) Moreover, the fractal oscillations of the equation $(P)$: $y''+f(x)y=0$, $x\in (0, 1]$, will be given, where $f(x)$ is positive and decreasing on $(0, 1]$, singular at $x=0$, that is $f(0+)=\infty$ and satisfies the so-called Hartman-Wintner condition near $x=0$, $f^{; ; -1/4}; ; (f^{; ; -1/4}; ; )''\in L^{; ; 1}; ; (0, 1)$ (see M. K. Kwong, M. Pa\v{; ; s}; ; i\'{; ; c}; ; , and J. S. W. Wong, 'Rectifiable Oscillations in Second Order Linear Differential Equations', J. Differential Equations, 245 (2008), 2333-2351). In the second part, we sketch the proofs of previous two main results on the fractal oscillations of linear differential equations. In order to prove that a function $y(x)$ is the fractal oscillatory on the interval $[0, 1]$, we need to find an $s\in [1, 2]$ such that $\dim_{; ; M}; ; G(y)=s$ and $0<M^{; ; s}; ; (G(y))<\infty $, where $G(y)$ is the graph of $y(x)$. According to the definitions of $\dim_{; ; M}; ; G(y)$ and $M^{; ; s}; ; (G(y))$ it is equivalent with the following precise asymptotics of the Lebesgue measure of the $\varepsilon $-neighborhood $G_{; ; \varepsilon }; ; (y)$ of $G(y)$: $|G_{; ; \varepsilon }; ; (y)| \sim\varepsilon^{; ; 2-s}; ; $, that is, $c_1\varepsilon^{; ; 2-s}; ; \leq |G_{; ; \varepsilon }; ; (y)|\leq c_2\varepsilon^{; ; 2-s}; ; $ for small $\varepsilon >0$ and for some positive constant $c_1$ and $c_2$. We show that the lower bound for $|G_{; ; \varepsilon }; ; (y)|$ ($c_1\varepsilon^{; ; 2-s}; ; \leq |G_{; ; \varepsilon }; ; (y)|$) is based on an easy geometric fact which allows us to inscribe a tringle into $G_{; ; \varepsilon }; ; (y)$ near $x=0$. However, in order to show the upper bound for $|G_{; ; \varepsilon }; ; (y)|$ ($|G_{; ; \varepsilon }; ; (y)|\leq c_2\varepsilon^{; ; 2-s}; ; $) there are many difficulties. Therefore, we present a short differential geometry of the $\varepsilon $-parallels of graph $G(y)$ which are essentially involved in the boundary of $|G_{; ; \varepsilon }; ; (y)|$. It also includes a study of two profiles of the curvature $\kappa_y(x)$: the cases when $\kappa_y(x)$ is a one-hump and a two-hump mapping on the interval of two consecutive inflexion-points of $y(x)$. The curvature $\kappa_y(x)$ influences the regularity and singularity of the $\varepsilon $-parallels of graph $G(y)$. Previous analysis of $\kappa_y(x)$ and the $\varepsilon $-parallels of graph $G(y)$ will be enlarged from a real function $y(x)$ to any solution $y(x)$ of linear differential equation. In the third part, some of previous results will be generalized to the half-linear equation $(|y'|^{; ; p-2}; ; y')'+f(x)|y|^{; ; p-2}; ; y=0$, $x\in (0, 1]$, $p>1$ (see M. Pa\v{; ; s}; ; i\'{; ; c}; ; and J. S. W. Wong, 'Rectifiable oscillations in second-order half-linear differential equations', Annali di matematica pura ed applicata, 188 (2009), 517-541). Also, some new ideas about the fractal oscillations in other types of differential equations will be proposed.

graph; box-dimension; Minkowski content; oscillations; ordinary differential equations

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