Summary form only given, as follows. Events of relative briefduration (involving for instance accelerated or heated particles,excitation of fluctuations, radiation emission) can be related to theonset of different kinds of explosive instabilities that can recur atregular intervals or randomly. An analytical model is introduced, torepresent these events consisting of a set of non-linear differentialequations which involve a characteristic singularity. This correspondsto an explosive or quasi-explosive event for a “primary”factor (e.g. the population of heated or accelerated particles) or forthe relevant plasma fluctuations that are excited when the primaryfactor exceeds an appropriate threshold value. In the case wherequasi-explosive events are periodically recurring a non-canonicalHamiltonian is derived from which the equations from both the primaryfactor and the excited fluctuation amplitude can be derived. Significantexamples of the numerical solutions of these equations are given. Acomparison is made with the well known Volterra-Lotka equations and withpreviously considered equations producing sawtooth oscillations of theprimary factor all of which do not involve singularities and do notdescribe explosive events. The random occurrence of this kind of events,involving the primary factor and the fluctuation level, is found byintroducing a relatively small time dependent component of the source ofthe driving factor or of the instability threshold for the fluctuationlevel, with a period that is not related to that of the originalnon-linear equations
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