engleski

Fractal properties of oscillatory solutions of a class of ordinary differential equations

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 od solutions of a class of ordinary differential equations, including Bessel equation. The phase dimension of solutions of Bessel equation has been computed to be equal to $4/3$. We also compute the phase dimension of a class of $(\alpha, 1)$-chirp-like functions, related to Bessel equation, to be equal to $2/(1+\alpha)$. We determined the box dimension of a specific type of spirals that we called \emph{;wavy spirals};, which are converging to the origin, but not with a nonincreasing radius function.

box dimension; oscillatory dimension; phase dimension; Bessel equation; wavy spiral

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