The uncertainty dimension and fractal boundaries for charged particle dynamics in the magnetotail

In this paper we examine the fractal nature of the basin boundary between forward scattered and backscattered particles as measured in the asymptotic region of the modified Harris model of the Earth magnetotail. It is shown that, in order to enter the chaotic region of phase space, an incoming ion (launched from above the midplane) must have an asymptotic pitch angle below a certain maximum value. This maximum pitch angle depends on the underlying structure of the phase space and takes on minimum (maximum) values at off-resonant (resonant) energies. Examples of the fractal basins are shown for both a resonant and off-resonant energy. Furthermore, we calculate the uncertainty exponent and the associated fractal dimension of the basin boundary as a function of the ion energy. We find that the uncertainty exponent takes on maximum (minimum) values at off-resonant (resonant) energies indicating that the box counting dimension of the basin boundary is farthest from integer values at the off-resonant energies. Finally, we show that in the integrable limit of vanishing normal component of the magnetic field, the uncertainty exponent approaches zero.