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This article discusses the application of mathematical models to capture emergent properties of complex physical systems. For other scientific and professional disciplines addressing complexity in their fields, see "complex systems.
A complex system is a "system composed of many components which may interact with each other. In many cases it is useful to represent such a system as a network where the nodes represent the components and the links their interactions. Examples of complex systems are Earth's global "climate, "organisms, the "human brain, social and economic organizations (like "cities), an "ecosystem, a living "cell, and ultimately the entire "universe.
Although it is arguable that humans have been studying complex systems for thousands of years, the modern scientific study of complex systems is relatively young in comparison to established fields of science such as "physics and "chemistry. The history of the scientific study of these systems follows several different research trends.
In the area of "mathematics, arguably the largest contribution to the study of complex systems was the discovery of "chaos in "deterministic systems, a feature of certain "dynamical systems that is strongly related to "nonlinearity. The study of "neural networks was also integral in advancing the mathematics needed to study complex systems.
The notion of "self-organizing systems is tied up to work in "nonequilibrium thermodynamics, including that pioneered by "chemist and "Nobel laureate "Ilya Prigogine in his study of "dissipative structures. Even older is the work by "Hartree-Fock c.s. on the quantum-chemistry equations and later calculations of the structure of molecules which can be regarded as one of the earliest examples of emergence and emergent wholes in science.
The first research institute focused on complex systems, the "Santa Fe Institute, was founded in 1984. Early Santa Fe Institute participants included physics Nobel laureates "Murray Gell-Mann and "Philip Anderson, economics Nobel laureate "Kenneth Arrow, and Manhattan Project scientists "George Cowan and "Herb Anderson. Today, there are over 50 "institutes and research centers focusing on complex systems.
The behaviour of non-linear systems is not subject to the principle of "superposition while that of "linear systems is subject to superposition. Thus, a complex "nonlinear system is one whose behaviour cannot be expressed as a sum of the behaviour of its parts (or of their multiples).
For a dynamical system to be classified as "chaotic, it must have the following properties:
minus the conjugate of z
, plus the original value of the pixel for each pixel, colouring pixels per the number iterations until the absolute value of z
exceeds two; display using inversion (borders are inner set), so that you can see that it threatens to fail the density condition, even if it meets the topological condition.
- it must be "sensitive to initial conditions,
- it must be "topologically mixing, and
- its "periodic orbits must be "dense.
Sensitivity to "initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future "trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.
Complex adaptive systems
"Complex adaptive systems (CAS) are special cases of complex systems. They are "complex in that they are diverse and made up of multiple interconnected elements and "adaptive in that they have the capacity to change and learn from experience. Examples of complex adaptive systems include the "stock market, social insect and "ant colonies, the "biosphere and the "ecosystem, the "brain and the "immune system, the "cell and the developing "embryo, "manufacturing businesses and any human social group-based endeavor in a cultural and "social system such as "political parties or "communities. This includes some large-scale online systems, such as collaborative tagging or "social bookmarking systems.
Complex systems may have the following features:
- "Cascading failures
- Due to the strong coupling between components in complex systems, a failure in one or more components can lead to cascading failures which may have catastrophic consequences on the functioning of the system.
Localized attack may lead to cascading failures in spatial networks.
- Complex systems may be open
- Complex systems are usually "open systems — that is, they exist in a "thermodynamic gradient and dissipate energy. In other words, complex systems are frequently far from energetic "equilibrium: but despite this flux, there may be pattern stability, see "synergetics.
- Complex systems may 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 spontaneous failures and recovery as well as "hysteresis.
Interacting systems may have complex hysteresis of many transitions.
- 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.
- 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 sufficiently determined by the activity of the systems' basic constituents, 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.
- 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 always 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.
- ^ History of Complex Systems
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