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).
- 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
- Ledford, H. (2015). How to solve the world's biggest problems. Nature, 525(7569), 308-311.
- Waldrop, M. M. (1993). Complexity: The emerging science at the edge of order and chaos. Simon and Schuster.
- EPSRC description of Non-linear systems retrieved 11 Aug 2015
- Hasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University Press. ISBN 0-521-58750-6.
- Alan Randall (2011). Risk and Precaution. Cambridge University Press. ISBN 9781139494793.
- S. V. Buldyrev; R. Parshani; G. Paul; H. E. Stanley; S. Havlin (2010). "Catastrophic cascade of failures in interdependent networks". Nature. 464 (7291): 08932. Bibcode:2010Natur.464.1025B. PMID 20393559. arXiv: . doi:10.1038/nature08932.
- Berezin, Yehiel; Bashan, Amir; Danziger, Michael M.; Li, Daqing; Havlin, Shlomo (2015). "Localized attacks on spatially embedded networks with dependencies". Scientific Reports. 5 (1). ISSN 2045-2322. doi:10.1038/srep08934.
- Majdandzic, Antonio; Podobnik, Boris; Buldyrev, Sergey V.; Kenett, Dror Y.; Havlin, Shlomo; Eugene Stanley, H. (2013). "Spontaneous recovery in dynamical networks". Nature Physics. 10 (1): 34–38. ISSN 1745-2473. doi:10.1038/nphys2819.
- Majdandzic, Antonio; Braunstein, Lidia A.; Curme, Chester; Vodenska, Irena; Levy-Carciente, Sary; Eugene Stanley, H.; Havlin, Shlomo (2016). "Multiple tipping points and optimal repairing in interacting networks". Nature Communications. 7: 10850. ISSN 2041-1723. doi:10.1038/ncomms10850.
- A. L. Barab´asi, R. Albert (2002). "Statistical mechanics of complex networks". Reviews of Modern Physics. 74: 47–94. Bibcode:2002RvMP...74...47A. arXiv: . doi:10.1103/RevModPhys.74.47.
- M. Newman (2010). Networks: An Introduction. Oxford University Press. ISBN 978-0-19-920665-0.
- Reuven Cohen, Shlomo Havlin (2010). Complex Networks: Structure, Robustness and Function. Cambridge University Press. ISBN 978-0-521-84156-6.
- Paolo Sibani & Henrik Jeldtoft Jensen (2013). Stochastic Dynamics of Complex Systems, ISBN 978-1-84816-993-7, World Scientific and Imperial College Press.
- Chu, Dominique (2011). Complexity: Against Systems. Theory in Biosciences, Springer Verlag. 
- Rocha, Luis M. (1999). "Complex Systems Modeling: Using Metaphors From Nature in Simulation and Scientific Models". BITS: Computer and Communications News. Computing, Information, and Communications Division. Los Alamos National Laboratory. November 1999
- Ignazio Licata & Ammar Sakaji (eds) (2008). Physics of Emergence and Organization, ISBN 978-981-277-994-6, World Scientific and Imperial College Press.
- Alfred Hübler, Cory Stephenson, Dave Lyon, Ryan Swindeman (2011). Fabrication and programming of large physically evolving networks Complexity, 16(5), pp. 7–8
- De Toni, Alberto; Comello, Luca (2011). Journey into Complexity. Udine: Lulu. ISBN 978-1-4452-6078-5.
- Introduction to complex systems-short course by Shlomo Havlin
- Complex systems in scholarpedia.
- (European) Complex Systems Society
- (Australian) Complex systems research network.
- Complex Systems Modeling based on Luis M. Rocha, 1999.
- CRM Complex systems research group
- The Center for Complex Systems Research, Univ. of Illinois at Urbana-Champaign
- FuturICT - Exploring and Managing our Future