Complex system

The collective name for systems consisting of a large number of interconnected elements. It must be emphasized that this is an informal concept, since there is yet no strict mathematical definition that embraces all intuitive notions of complex systems. Typical examples of complex systems are: the nervous system, the brain, a computer, a control system in human society, etc.

In the 20th century, the need to investigate objects of ever-increasing complexity has invoked the introduction of complex models in many branches of science: biology, technology, economics, and sociology, among others. In particular, cybernetics arose as an independent science. In it a fundamental notion is that of a complex control system. As a result, a number of special disciplines have also arisen, including the word "system" in their names, such as systems analysis, system-technology, and general systems theory, among others.

There are various mathematical approaches to the description and the study of complex systems. Two types of mathematical models of complex systems can be identified, the discrete and the continuous. The former are studied mainly in mathematical cybernetics (the theory of control systems) and rely on the apparatus of discrete mathematics, while the latter are studied in the theory of dynamical systems (cf. Dynamical system) and in automatic control theory, and have as mathematical basis the theory of differential equations. Also widely used in the study of complex systems are probabilistic-statistical methods and models, for example in scheduling theory, stochastic programming and stochastic simulation. Despite the difference in form and mathematical apparatus used, all these approaches to the description of complex systems are united by a common methodology and a common object of study.

One of the most difficult points in all attempts to give a mathematical description of complex systems is the formalization of the concept of complexity. Many of the characteristic features of "complexity" are inherent in actual complex systems: a large number of elements, a variety of possible forms of relations between them, complex functioning, hierarchy of structure, etc. Note that the terms complex system and "large (scale) system" are not synonymous, since the latter embraces systems with only one complex feature: a large number of elements.

Basic advances in the formalization of the concept of complexity in the mathematical study of complex systems have recently (1983) been achieved for sufficiently simple (model) classes of control systems, such as Turing machines, diagrams of functional elements, finite automata, etc. (cf. Turing machine; Diagram of functional elements; Automaton, finite). Current interest centers on the study of increasingly complex mathematical models, with a view to reflecting more fully the structure and functioning of actual complex systems. Many regular features of simple models can be extended to more complex ones.

References

[1]	A.A. Lyapunov, S.V. Yablonskii, Problemy Kibernetiki , 9 (1963) pp. 5–22
[2]	N.I. Buslenko, V.V. Kalashnikov, I.N. Kovalenko, "Lectures on the theory of complex systems" , Moscow (1973) (In Russian)
[3]	, Encyclopaedia of cybernetics , 2 , Kiev (1975) pp. 373–375 (In Russian)