Recurrent and random explosive events: Relevant theoretical models

Summary form only given, as follows. Events of relative brief duration (involving for instance accelerated or heated particles, excitation of fluctuations, radiation emission) can be related to the onset of different kinds of explosive instabilities 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 "primary" 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 from both the primary factor and the excited fluctuation amplitude can be derived. Significant examples of the numerical solutions 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 equations.