An analysis of chaotic confinement regions described by a charged particle subjected to the magnetic field of a dipole

This paper aimed to make an analysis of the confinement regions described by a charged particle subjected to the magnetic field of a dipole. The equations of motion were obtained and analyzed, from which it was possible to demonstrate that the mechanical energy of the system is constant. The obtained confined trajectories are described by the particle around the z axis and maintaining a certain minimum distance Δr > 0 from the origin (where the source is located), which is smaller the smaller the values of the initial conditions in the xy plane, with the limit that these values must be larger than the dimensions of the source. Moreover, an analysis of the system was performed in the particular case z(t) = 0, in which the non-existence of chaos was verified by the Arnold- Lioiville theorem and becomes chaotic when the dipole moment is recalibrated with oscillating time dependence, and general z(t), in which chaos and a transient behavior of the maximum Lyapunov exponents as a function of the number of steps of the numerical calculation was verified. The confinements treated in this paper represent possible approximations for phenomena that happen in the atmosphere of planet Earth.

Keywords
Charged particle trapping; Chaos; Poincaré sections and Lyapunov exponents