engleski

Fractal properties of generalized Bessel functions

The fractal oscillatority of solutions $x=x(t)$ of ordinary differential equations at $t=\infty$ is measured by \emph{; ; oscillatory}; ; and \emph{; ; phase dimensions}; ; defined through the box dimension. Oscillatory and phase dimensions are defined as box dimensions of the graph of $X(\tau)=x(1/\tau)$ near $\tau=0$ and trajectory $(x, \dot{; ; x}; ; )$ in $\R^2$, 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 oscillatory dimension of solutions of Bessel equation has been determined by Pa\v si\'c and Tanaka (2011). Here, we compute the phase dimension of solutions of a class of ordinary differential equations, including Bessel equation. These solutions we call generalized Bessel functions. The phase dimension of Bessel functions is computed to be equal to $4/3$. We relate these results to the result from \v Zubrini\'c and \v Zupanovi\'c (2008) about the box dimension of spiral trajectories of planar vector fields, depending on the asymptotic behavior of iterates of the Poincar\'{; ; e}; ; map. They applied it to the Hopf bifurcation and Li\'{; ; e}; ; nard systems. Also, they obtained all possible values of box dimensions of spiral trajectories around a weak focus, associated with polynomial vector fields. Computation of the phase dimension of generalized Bessel functions use asymptotic expansions of Bessel functions. Due to a very large number of terms, we have to employ methods of computer algebra.

box dimension; oscillatory dimension; phase dimension; Bessel equation

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