**4.6. The principle of critical energy, synergy and catastrophe theory**

The principle of critical energy combines the essentials of synergetics and catastrophe theory. This statement is easy to account for:


Such a sudden transformation has been called catastrophe [38]. It underlies the theory of catastrophe [39]. Here are some examples of leap-type transformations which are the object of the theory of catastrophes and also lie at the core of some particular cases of PCE application: buckling bars, ice melting, water boiling, earthquakes, the camel back likely to withstand *n* loadings, but fails—as well known—under load *n* + 1, a cell that suddenly changes its repro‐ duction rate, doubles and redoubles, etc. Some phenomena are triggered when the friction forces are overcome, like the rustling of a plant, or the noise of an earthquake.

The cumulation of the state of stress (frustration, desolation), coupled with alienation (alien‐ ation, lack of communication) may lead at a certain moment, in prisons, to rebellion. There is a sudden violent switch from quiet to disturbance, to disorder.

Bar buckling, for example, was first analysed by Euler [40] that is so long before the emergence of the catastrophe theory, a theory capable to bring some clarity on structural instability,

<sup>1</sup> Synergetics studies: the common actions of the micro-components of an open system meant to attain a certain goal. With this aim in view, the system may exchange energy and mass; the macroscopic structures that appear in the sudden transition between the two states of the system, due to their cooperation.

For instance, when a **critical value** of a physcial parametre has been exceeded, the microscopic components of the system may suddenly start working in the same direction. The effect obtained at a macroscoppic level is a state of the system that is qualitatively different from the previous state. [37] is precisely the cooperation principle (the principle of effects superposition through self-organizing at a microscopic scale).

characterized by abrupt changes when the critical value of loading is reached. The problem of structural stability (including bar buckling) was extensively analysed on the basis of the principle of critical energy in [41].
