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There are many definitions of complexity, therefore many natural, artificial and abstract objects or networks can be considered to be complex systems, and their study (complexity science) is highly interdisciplinary. Examples of complex systems include ant-hills, ants themselves, human economies, nervous systems, cells and living things, including human beings, as well as modern energy or telecommunication infrastructures.

Beyond the fact that these things are all networks of some kind, and that they are complex, it may appear that they have little in common, hence that the term "complex system" is vacuous. However, all complex systems are held to have behavioural and structural features in common, which at least to some degree unites them as phenomena. They are also united theoretically, because all these systems may, in principle, be modelled with varying degrees of success by a certain kind of mathematics. It is therefore possible to state clearly what it is that these systems are supposed to have in common with each other, in relatively formal terms.

Definition

The term complex system has no precise definition but can often be taken to mean a system with many strongly-coupled degrees of freedom. Traditional methods of mathematical modeling are adept at handling systems with few degrees of freedom that interact strongly, such as the paradigmatic simple harmonic oscillator, while statistical methods are useful for systems with very many degrees of freedom all of which interact weakly, such as a box of gas. Complex systems such as the Potts model occupy the intermediary regime, where local and global phenomena interact in complicated, often nonlinear ways. (For consistency, the examples cited above are all drawn from physics even though complex systems are studied by many scientific disciplines.)

Applications of complex systems theory

The study of complex systems is bringing new vitality to many areas of science where a more typical reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including neuroscience, meteorology, chemistry, physics, computer science, psychology, artificial life, evolutionary computation, economics, earthquake prediction, heart cell synchronisation, immune systems, reaction-diffusion systems, molecular biology, epilepsy and inquiries into the nature of living cells themselves. In these endeavours, scientists often seek simple non-linear coupling rules which lead to complex phenomena (rather than describe - see above), but this need not be the case. Human societies (and probably human brains) are complex systems in which neither the components nor the couplings are simple. Nevertheless, they exhibit many of the hallmarks of complex systems.

Traditionally, engineering has striven to keep its systems linear, because that makes them simpler to build and to predict. However, many physical systems (for example lasers) are inherently "complex systems" in terms of the definition above, and engineering practice must now include elements of complex systems research.

Information theory applies well to the complex adaptive systems, CAS, through the concepts of object oriented design.

Features of complex systems in nature

Relationships are non-linear

In practical terms, this means a small perturbation may cause a large effect (see butterfly effect), a proportional effect, or even no effect at all. In linear systems, effect is always directly proportional to cause. See nonlinearity.

Relationships contain feedback loops

Both negative (damping) and positive (amplifying) feedback are often found in complex systems. The effects of an element's behaviour are fed back to in such a way that the element itself is altered.

Complex systems are open

Complex systems in nature are usually open systems — that is, they exist in a thermodynamic gradient and dissipate energy. In other words, complex systems are usually far from energetic equilibrium: but despite this flux, there may be pattern stability. See synergetics.

Complex systems have a memory

The history of a complex system may be important. Because complex systems are dynamical systems they change over time, and prior states may have an influence on present states. More formally, complex systems often exhibit hysteresis.

Complex systems may be nested

The components of a complex system may themselves be complex systems. For example, an economy is made up of organisations, which are made up of people, which are made up of cells - all of which are complex systems.

Boundaries are difficult to determine

It can be difficult to determine the boundaries of a complex system. The decision is ultimately made by the observer.

Dynamic network of multiplicity

As well as coupling rules, the dynamic network of a complex system is important. Small-world or scale-free networks which have many local interactions and a smaller number of inter-area connections are often employed. Natural complex systems often exhibit such topologies. In the human cortex for example, we see dense local connectivity and a few very long axon projections between regions inside the cortex and to other brain regions.

May produce emergent phenomena

Complex systems may exhibit behaviors that are emergent, which is to say that while the results may be deterministic, they may have properties that can only be studied at a higher level. For example, the termites in a mound have physiology, biochemistry and biological development that are at one level of analysis, but their social behavior and mound building is a property that emerges from the collection of termites and needs to be analysed at a different level.

Quotes

From Sync by Steven Strogatz: "Every decade or so, a grandiose theory comes along, bearing similar aspirations and often brandishing an ominous-sounding C-name. In the 1960 it was cybernetics. In the '70s it was catastrophe theory. Then came chaos theory in the '80s and complexity theory in the '90s."

Various informal descriptions of complex systems have been put forward, and these may give some insight into their properties. A special edition of Science about complex systems Science Vol. 284. No. 5411 (1999). highlighted several of these:

A complex system is a highly structured system, which shows structure with variations (Goldenfeld and Kadanoff)

A complex system is one whose evolution is very sensitive to initial conditions or to small perturbations, one in which the number of independent interacting components is large, or one in which there are multiple pathways by which the system can evolve (Whitesides and Ismagilov)

A complex system is one that by design or function or both is difficult to understand and verify (Weng, Bhalla and Iyengar)

A complex system is one in which there are multiple interactions between many different components (D. Rind)
Complex systems are systems in process that constantly evolve and unfold over time (W. Brian Arthur).

External links

Institutes

Directories

Articles

Supplements to the Proceedings of the National Academy of Sciences (PNAS):
- Adaptive Agents, Intelligence, and Emergent Human Organization: Capturing Complexity through Agent-Based Modeling
- Self-organized complexity in the physical, biological, and social sciences
CSCS's Definition of Complex Systems
NICO's About Complex Systems
Complex networks - augmenting the framework for the study of complex system (Amaral & Ottino, 2004)
Self-Organizing and Self-Replicating Paths to Autonomous Intelligence (A.I.)
A New Kind of Science by Stephen Wolfram
Plectics article from by Murray Gell-Mann
Emergent Nature
Simple Lessons from Complexity (May not be available)
VisualComplexity.com - A visual exploration on mapping complex networks
Modelling Complex Socio-Technical Systems using Morphological Analysis From the Swedish Morphological Society
Theory of strongly correlated systems
Complexity; a science at 30
Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth

Mailing Lists, Discussion Groups and Forums

Specialized Wikis and Sites about Complex Systems

Conferences

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 * [[Dynamical system]]
 * [[Nonlinearity]]
+* [[Simulated reality]]
 * [[Systems theory]]
 * [[Multi-agent system]]