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Abrupt phenomena in modelling real-world systems indicate the importance of investigating systems with steep gradients.
Unfortunately, the stochastic disturbances in real world do not obey Gaussian noise, but they are variable structures subject to stochastic abrupt changes, which may come from abrupt phenomena.
For this case, it is recognized that stochastic differential equations with Lévy noise are quite suitable to describe such stochastic abrupt phenomena.
A lot of difference systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc.
Markovian jump systems (MJSs) involve both time-evolving and event-driven mechanisms, which can be employed to model abrupt phenomena such as random failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc.
For networks with communication constraints, such random occurring coupling may be subject to random abrupt changes, which may have resulted from abrupt phenomena such as random failures, changes in the interconnections of subsystems and sudden environment changes, and so forth.
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There is also evidence that the nonseizure (inter-ictal) to seizure (ictal) transition is not an abrupt phenomenon [29].
The late Early to Middle Pleistocene terrestrial faunal "revolution" actually was not an "abrupt" phenomenon, which started from about 1.3 Ma giving rise to a progressive reconstruction of mammalian faunal complexes, which ended during the early Ionian.
It is unclear whether these assumptions apply to the trial, and more widely to clinical practice, since loss of consciousness is an abrupt phenomenon during induction of anaesthesia.
In order to describe the dynamics of populations subject to abrupt changes and other phenomena such as harvesting, diseases, and so on, some authors have used an impulsive differential system to describe these kinds of phenomena since the last century.
Recently, a new type of neural networks--impulsive neural networks display a combination of characteristics of both the continuous-time and discrete-time systems, which is an appropriate description of the phenomena of abrupt qualitative dynamical changes of essentially continuous-time systems, see [4, 9, 13 22].
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