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Translations of Encyclopedia about Mathematics

 

Dynamic Systems

An example of a dynamic system is a flowing liquid, turbulence, a biological ecosystem or a meteorological model. Such systems often react immensely or unpredictably to the slightest error. Even though such systems do not operate randomly, every flow is influenced by certain factors (heat, height, air dampness, the soilís surface). However, these factors are so various and their affect so complex that the long term development of such a system is not easily forecastable because, for exact prediction, all factors with their mutual interaction must be known. In fact, this ideal example has a name: Laplaceís demon.

If we assumed there existed a demon capable of controlling all factors influencing the weather and capable of calculating the various factors, at any point, this demon would be capable of predicting the weather. In short, this means that what takes place within a complex system is not random but rather the result of definite conditions (the relation between cause and consequence applies here). However, from a certain point, these causal relationships remain inaccessible to us.

For dynamic systems, time and the resulting, constant change, is also a characteristic factor. The quantities of an open, dynamic system (such as when a tree exchanges matter with its surroundings) determine its behaviour. All other system quantities can then be derived from this, although the external influencing factors and consequences of various events lead to changes in the conditional quantities.

Examples of changes which may take place over a longer period of time are growth, dissolution, oscillation or stabilisation.

For example, such a tree can be a concrete example of a dynamic system in that it grows, develops its structure and withers away.

System signs using the tree example:

  1. A defined goal must be fulfilled.
    A tree grows and plants seeds.
  2. Elements of the system must be interconnected in their effect.
    Elements of the system, such as the leaves, branches, trunk and roots, are mutually interconnected.
  3. The system cannot be broken apart.

If we were to remove the tree, some of the system elements could not serve their purpose.

The structure of the system is made up of a defined number of system components and the relations between these components. Complex systems have many parts, between which there exists a thick network of relations.

The behaviour or the dynamics of the system is very variable. Complex behaviour often, but not always, depends on past states. Occurrences in a running state are often irreversible. Short term predictions are possible in the behaviour of complex systems, but not long term predictions. A simple example of this is meteorological forecasts. Complex behaviours cannot easily be simplified because they contain a lot of information and because their patterns rarely repeat.

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