Delay coordinates: a sensitive indicator of nonlinear dynamics in single charged particle motion in magnetic reversals

The delay coordinate technique is examined as an indicator of the regime of particle dynamics for the system of single charged particle motion in magnetic reversals. Examples of numerically integrated trajectories in both static (zero electric field) and time dependent (corresponding nonzero induction electric field) simple models for magnetic reversals are considered. In the static case, the dynamics can in principle be directly classified by constructing Poincaré surfaces of section; here we demonstrate that whilst the Poincaré surface contains the relevant information to classify the dynamics, the corresponding delay coordinate plot can provide a far more sensitive indication of the onset of nonregular behaviour. In the case of nonperiodic time dependence considered here Poincaré plots cannot in general be constructed directly. Nevertheless, delay coordinate plots can still reveal details of the phase space portrait of the system, and here are shown to indicate whether segments of stochastic motion exist in a given trajectory. It is anticipated that the delay coordinate plot technique as realized here will be a valuable tool in characterizing the behaviour in large numbers of trajectories that are evolved in time-dependent systems, thereby giving us insight into the evolution of the distribution function as a whole, either in prescribed fields or in self-consistent numerical simulations.