Uncertainty exponents for chaotic particle scattering in the magnetotail

In chaotic scattering systems such as the magnetotial, small uncertainties in the initial conditions make it difficult to predict whether a given particle will be forward-scattered or back-scattered in its interaction with the field reversal. A particle with initial conditions x0, is said to be ɛ certain if all of the particles with initial conditions x such that |x - x0|<ɛ are scattered in the same direction. For a simple non-fractal boundary between the two scattering basins of attraction, the fraction of initial conditions f(ɛ) is proportional to ɛ. For fractal basin boundaries, f(ɛ) is proportional to ɛ^α where α=N-D0 is the uncertainty exponent, N is the dimension of the phase space and D0 is the box counting dimension of the basin boundary. For α<1, a decrease in f(ɛ) by a factor of 10 mandates a reduction in ɛ by a factor of 10^(1/α) and thus there is a high level of sensitivity to the initial conditions, (a hallmark of chaos.) In this paper we have examine the uncertainty exponent for chaotic particles in the modified Harris model of the magnetotail as a function of both the energy and the normal component of the magnetic field. For each parameter set, we launching 1000 chaotic orbits from the asymptotic region and then follow four additional orbits whose initial conditions vary by ɛ. If all 5 initial conditions scatter in the same direction, the particle is ɛ certain. We then vary ɛ over 7 order of magnitude and determine f(ɛ) as a function of ɛ. We find that the value of α takes on minimum values near discrete resonant energies and that as the normal component of the magnetic field goes to zero, α also goes to zero.