International Encyclopedia of Systems and Cybernetics

The theory of oscillating processes and systems has been developed in succesive steps, its models progressively nearing the concrete situations.

The simplest oscillator shows a regular and invariable periodic behavior. It does not take into account the degradation of energy which results, for example, of frictions. This type of oscillators is neither time-dependent nor environment- dependent and can be described by any function characteristic of non-dissipative processes. It has no attractor.

However, such a model is totally abstract and theoretical, in close relation to the concept of isolated system, in use in classical thermodynamics.

Conversely the next model of oscillator considers the irreversible dissipation of energy and growth of entropy. In this case, more realistic, the system's trajectory tends to a final point of static stability, i.e. a point of attraction, or simply, an attractor.

This is the case, for example of a pendulum that receives no supplementary energy from its environment and is thus not able to compensate for the monotonous loss of usable potential. The behavior of this class of oscillators is defined by a positive and monotonously decreasing LYAPOUNOV's function, corresponding to the progressive damping of the amplitude of their movements.

There are however numerous systems which cannot be represented by such a model: those which receive from their environment an energy supply equal or superior to the energy that they able to dissipate. In the first of these cases, the system heightens its entropy production up to a level at which it becomes able to maintain its dynamic stability for a long time. (Equilibrium thermodynamics). This case corresponds to the emergence of a stable limit cycle characterized by a VAN DER POL's attractor.

If the energy input becomes superior to the dissipation capacity of the system, it will undergo uncontrolable growing fluctuations, eventually cross a threshold, become unstable and pass through bifurcation points corresponding to deep alterations. This may lead to chaotic behavior and the future of the oscillating system becomes unpredictable.