Recurrent and random explosive events: Relevant theoretical models

There are events of relatively brief duration (involving for instance accelerated or heated particles, excitation of fluctuations, radiation emission) that can be related to the onset of different kinds of explosive instabilities and that can recur at regular intervals or randomly. An analytical model is introduced, to represent these events, consisting of a set of non-linear differential equations which involve a characteristic singularity. This corresponds to an explosive or quasi-explosive event for a open-quotes primaryclose quotes factor (e.g. the population of heated or accelerated particles) or for the relevant plasma fluctuations that are excited when the primary factor exceeds an appropriate threshold value. In the case where quasi-explosive events are periodically recurring a non-canonical Hamiltonian is derived from which the equations for both the primary factor and the excited fluctuation amplitude can be derived. Significant examples of the numerical solution of these equations are given. A comparison is made with the well known Volterra-Lotka equations and with previously considered equations producing sawtooth oscillations of the primary factor all of which do not involve singularities and do not describe explosive events. The random occurrence of this kind of events, involving the primary factor and the fluctuation level, is found by introducing a relatively small time dependent component of the source of the driving factor or of the instability threshold for the fluctuation level, with a period that is not related to that of the original non-linear equation. This case is also illustrated by representative numerical solutions