The Lyapunov exponent as a measure of chaos in the magnetotail

In this paper we re-examine chaotic nature of charged particle motion in the magnetotail focusing on the:1) the Lyapunov exponent (λ) 2) the fraction of the orbits that enter into the chaotic region of phase space and 3) the average trapping time of the orbits. The Lyapunov exponent is calculated by finding the time asymptotic value of a distribution of λ’s for a uniform source distribution of 500,000 particles. We show that at particular resonance energies the number of particles entering into the chaotic region of phase space experiences a minimum and that both the Lyapunov exponent and the time that the particles are trapped have maxima. Furthermore, we find that the longest lived orbits at the resonance energies uniformly sample the chaotic region of phase space, whereas the longest lived orbits at off resonant energies are “sticky” orbits that are trapped near the integrable regions of phase space and only sparsely sample the chaotic regions. As a general trend, the value of λ tends to increase for increasing values of the energy. It is shown that as the normal component of the magnetic field approaches zero (i.e. in the limit of a completely integrable system), λ also approaches zero. These results are in good agreement with a simple phenomenological model for the Lyapunov exponent. Finally, we have defined a “chaos” parameter as the product of the Lyapunov exponent time the fraction of particles that enter into the chaotic region times the average trapping time. It has been conjectured that the ``chaos'' should be a maximum when the ion gyroradius is equal to the maximum curvature of the magnetic field. Interestingly, whereas neither the Lyapunov exponent nor the trapping time is a maximum near this energy, both the fraction of orbits entering the chaotic region and the chaos parameter are at a maximum.