Naslov
Fractal analysis of oscillatory solutions of a class of ordinary differential equations including the Bessel equation

Autori
Domagoj Vlah

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Skup
Theoretical and computational methods in dynamical systems and fractal geometry

Mjesto i datum
Maribor, Slovenija, 07.04.2015. - 11.04.2015

Vrsta sudjelovanja
Pozvano predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
fractal dimension; box dimension; chirp-like function; Bessel function; Fresnel integral

Sažetak
In this talk we investigate oscillatority of functions using the fractal dimension. We apply this approach to some common objects of interest in that subject. These objects that we investigate, are chirp-like functions, Bessel functions, Fresnel oscillatory integrals and some generalizations. We first, from the point of view of fractal geometry, study oscillatority of a class of real $C^1$ functions $x=x(t)$ near $t=\infty$. A fractal oscillatority of solutions of second-order differential equations near infinity is measured by \emph{; ; oscillatory}; ; and \emph{; ; phase dimensions}; ; , defined as box dimensions of the graph of $X(\tau)=x(\frac{; ; 1}; ; {; ; \tau}; ; )$ near $\tau=0$ and trajectory $(x, \dot{; ; x}; ; )$ in $\mathbb{; ; R}; ; ^2$, respectively, assuming that $(x, \dot{; ; x}; ; )$ is a spiral converging to the origin. The box dimension of a plane curve measures the accumulation of a curve near a point, which is in particular interesting for non-rectifiable curves. The phase dimension has been calculated for a class of this oscillatory functions using formulas for box dimension of a class of nonrectifiable spirals. Also, the case of rectifiable spirals have been studied. A specific type of spirals that we called \emph{; ; wavy spirals}; ; , converging to the origin, but with an increasing radius function in some parts, emerged in our study of phase portraits. We further study the phase dimension of a class of second-order nonautonomous differential equations with oscillatory solutions including the Bessel equation. We prove that the phase dimension of Bessel functions is equal to $\frac{; ; 4}; ; {; ; 3}; ; $, and that the corresponding trajectory is a wavy spiral, exhibiting an interesting behavior. The phase dimension of that specific generalization of the Bessel equation has been also computed. Then we study some other class of second-order nonautonomous differential equations, and the corresponding planar and spatial systems, again from the point of view of fractal geometry. Using the phase dimension of a solution of the second-order equation we compute the box dimension of a spiral trajectory of the corresponding spatial system, lying in Lipschitzian or H\" olderian surfaces. This phase dimension of the second-order equation is connected to the asymptotics of the associated Poincar\'e map. Finally, we obtain a new asymptotic expansion of generalized Fresnel integrals $x(t)=\int_0^t\cos q(s)\, ds$ for large $t$, where $q(s)\sim s^p$ when $s\to\infty$, and $p>1$. The terms of the expansion are defined via a simple iterative algorithm. Using this we show that the box dimension of the related $q$-clothoid, also called the generalized Euler or Cornu spiral, is equal to $d=2p/(2p-1)$. This generalized Euler spiral is defined by generalized Fresnel integrals, as component functions, where $x(t)$ is as before and $y(t)=\int_0^t\sin q(s)\, ds$. Furthermore, this curve is Minkowski measurable, and we compute its $d$-dimensional Minkowski content.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA