The following text is a renewed version of two published articles (Crevecoeur,1993,1994).

A system is defined as a group of many interlocking parts operating together with a common goal (after Forrester, 1968: "many" was added). This is a different definition than the usual, when people speak of systems of two components etc. Because of the many elements, some kind of "complexity" appears and one will rather call such systems "complex systems". A "macro-system" is defined by the huge difference of scale between the system and its unitary components: say factor one to 1 billion, or to 1 billion of billions or Avogrado's number. The unitary components may be assembled in subsystems. For instance, a piece of metal composed of atoms (assembled in grains), a cell composed of molecules (assembled in subunits of different functionalities), a body composed of cells (assembled in organs and subsystems) and an universe composed of stars (assembled in galaxies) are macro-systems. Here we will speak only of macro-systems and simply call them "systems". In order to be precise, we will define the class of systems we are studying as follows:

A very simple way to understand what happens is the following. The operating of the system results in internal challenges on the subparts which may jeopardize the system's integrity. The subparts have to adapt to the challenges in due time in order to allow further operation of the system as a whole. At each time, each subpart will modulate its available operating resources in a way to give the most appropriate response to the particular challenge it has to meet. The corresponding mathematical expression is given by Equation (1).

However, during operation, the challenges are not removed: they appear again and again, sometimes on a steady base sometimes not, with given kinetics depending on the operating constraints. It may happen that because of the kinetics or because of the kind of operation constraints, the adaptations in the subparts be sometimes non-standard even if satisfactory for the further operation of the system. The word "non-standard" means "different from what could be expected from the past behaviour". This may result in changes of information for neighbouring subparts about what their challenges are. Their responses could then also become non-standard. There will be a snowball effect: non-standard responses will induce differing assessments of the challenges which in turn will induce other non-standard responses. This corresponds to positive first order feedback loops as showed in Figure 2. The corresponding mathematical expression is an increasing exponential: see Equation (2). Indeed, supposing the increase in the number "z(t)" of non-standard responses at any time to be proportional to this number (i.e., existing non-standard responses are systematic sources of new non-standard response), one has : dz(t)/dt=a .z(t) - with : a = constant - which becomes Equation (2) after integration. Non-standard responses have no special value for the system. Sometimes, they will improve the system's organization, sometimes, they will hamper it locally.

The evolution of the system as a whole with time will be the resultant of multiple combinations of first order negative feedback loops (corresponding to local adaptive responses) and positive feedback loops (corresponding to local non-standard responses) on subparts.

Now, we have to take into account that the system is, by definition, in a state of self-organized criticality. Therefore, the intervals between adaptations will be statistically distributed according to a power law. This will be reflected by a decrease of parameter "

The combination of multiple negative and positive first order feedback loops taking account of self-organized criticality will lead to global evolution curves of the kind shown on Figure 3. This can be illustrated on a simple example. The resulting ageing kinetics of the system as a whole appears to follow three steps: (1) decrease - (2) near constancy - (3) increase of the ageing rate. The first step corresponds to what is usually observed as "learning". Many observations point to the fact that these three steps are always the same and in the same sequence.

It must be emphasized that the kind of curve given in Figure 3 is the envelope of the many internal processes which at each time increment combine negative and positive first order feedback loops on subparts.

Now, zooming onto small parts of the curve of Figure 3 will show irregular shapes as illustrated on Figure 3-1 for regions in the beginning and Figure 3-2 for regions at the end of the curve. The actual shape will depend on the particular set of combinations of negative and positive feedback loops for the system under consideration (see alternative values of parameters in the example).

The typical 3-step curve of Figure 3 is well fitted by following equation:

Indeed, the envelope curve of adaptation curves as given by Equation (1) taking account of self-organized criticality in time, will be a power law with an exponent less than unity.

When above equation is derived in function of time (expressed by a point "." above the parameter), one finds:

As will be illustrated in chapter 5, these two equations are basic to describe the ageing or evolution kinetics of systems. The reverse of parameter "

Moreover, we see that systems following above equations will behave between two extremes:

(1) for

(2) for

These two types of behaviour, polynomial and exponential, are encountered in several fields: reliability of systems, cosmology and biology.

Now let us further analyze the two equations of evolution. The three parameters

(1) The system has to operate under given conditions. In function of these conditions, the challenges on the subparts will vary of intensity. Severe operating conditions will induce challenges of higher intensity than mild operating conditions.

(2) Moreover, the intensity of the challenges on each subpart will also depend on the internal organization of the system. A better organization will allow better adaptations at each time increment and lower the risk that non-standard responses be given to the challenges.

(3) Finally, physics commands that ageing or evolution processes be temperature dependent (Arrhenius plots should be observed).

Let us call these three constraints: (1)the "stress

The particular form of these equations must be determined for each class of systems, for instance experimentally by fixing two constraints and letting the third one vary. In some cases, this is already known thanks to extensive published test results (see, for instance, results of creep tests). Let us emphasize the meaning of the "stress

Now, as already mentioned, the reverse of parameter

Equation E-1 can then be re-written as Equation E-6. After t

Two kinds of predictions can be
performed using the mathematical model of system evolution kinetics: (1) for a given system
in operation ; (2) for a class of systems when the behaviour of several systems
of the class is known. Of course, it is, at this stage, impossible to follow the
particular responses of each subpart to its challenges when time proceeds.
Therefore, it is not that kind of prediction which will be made that deduces the
behaviour of the system from the knowledge of the behaviour of its subparts. It
must be emphasized again that the equations used here reflect the **global** behaviour of the system no matter what happens
precisely in the subparts: this global behaviour is the integration of all
internal adaptations.

Equations E-1 & E-2 can be used.

The ** first **thing to find out is which measurable
parameter is the best one to reflect the global ageing or evolution of the
system under scope. Often, several parameters are available. Experience teaches
us which parameters can be used. Examples of such parameters are (see also the
examples in Chapter 5): a
cumulative number of failures, a growth parameter, any parameter known as reflecting
variations in ageing level, like: resistance, force, strength, rapidity
of reaction, deformation, etc. as they are defined in the different sciences. We
will call this parameter the "evolution parameter" - E(t) in Equation E-1.

The

The

When these three points are defined, the evolution parameter must be measured at successive time intervals significant with regard to the timescale (eg. tenths or hundredths of the timescale). The values of

As already seen, when critical points exist for systems, an alternative equation can be used to predict them. One just takes Equation E-1 rewritten in normalized form:

Again, it is sufficient to record the values of the evolution parameter at successive time intervals to find "E

It is noteworthy that parameters of Equations E-1 or E-6 are not defined from the beginning. They are fixed progressively with time as the adaptations proceed. No system will exactly behave as another one. There are small local differences in stress, temperature and organization and, even if they occur at levels several orders of magnitude below that of the system operating as a whole, due to the kinetics of the adjustments and the irreversibility of time, the differences will accumulate resulting in different evolution or ageing histories. However, after a time, the values of

Mechanical systems reliability

Creep

Biology and Darwinian evolution

Cosmology

More fields of application might be found in the future. Let us anyway quote some connexions to theoretical concepts (self-organized criticality, Boltzmann's entropy, minimum entropy production theorem of Prigogine, etc.) in following item:

Statistical thermodynamics

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