Complex systems present problems both in mathematical modelling and philosophical foundations. The study of complex systems represents a new approach to science that investigates how relationships between parts give rise to the collective behaviors of a system and how the system interacts and forms relationships with its environment.
The equations from which models of complex systems are developed generally derive from statistical physics, information theory and non-linear dynamics, and represent organized but unpredictable behaviors of natural systems that are considered fundamentally complex. The physical manifestations of such systems are difficult to define, so a common choice is to identify "the system" with the mathematical information model rather than referring to the undefined physical subject the model represents. One of a variety of journals using this approach to complexity is Complex Systems.
Such systems are used to model processes in computer science, biology, economics, physics, chemistry, and many other fields. It is also called complex systems theory, complexity science, study of complex systems, sciences of complexity, non-equilibrium physics, and historical physics. A variety of abstract theoretical complex systems is studied as a field of mathematics.
The key problems of complex systems are difficulties with their formal modelling and simulation. From such a perspective, in different research contexts complex systems are defined on the basis of their different attributes. Since all complex systems have many interconnected components, the science of networks and network theory are important aspects of the study of complex systems. A consensus regarding a single universal definition of complex system does not yet exist.
For systems that are less usefully represented with equations various other kinds of narratives and methods for identifying, exploring, designing and interacting with complex systems are used.
The study of mathematical complex system models is used for many scientific questions poorly suited to the traditional mechanistic conception provided by science. Complex systems is therefore often used as a broad term encompassing a research approach to problems in many diverse disciplines including anthropology, artificial intelligence, artificial life, chemistry, computer science, economics, evolutionary computation, earthquake prediction, meteorology, molecular biology, neuroscience, physics, psychology and sociology.
In these endeavors, 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. It is worth remarking that non-linearity is not a necessary feature of complex systems modeling: macro-analyses that concern unstable equilibrium and evolution processes of certain biological/social/economic systems can usefully be carried out also by sets of linear equations, which do nevertheless entail reciprocal dependence between variable parameters.
Traditionally, engineering has striven to solve the non-linear system problem while bearing in mind that for small perturbations, most non-linear systems can be approximated with linear systems significantly simplifying the analysis. Linear systems represent the main class of systems for which general techniques for stability control and analysis exist. 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, as well as through formalized concepts of organization and disorder that can be associated with any systems evolution process.
Complex systems is a new approach to science that studies how relationships between parts give rise to the collective behaviors of a system and how the system interacts and forms relationships with its environment.
The earliest precursor to modern complex systems theory can be found in the classical political economy of the Scottish Enlightenment, later developed by the Austrian school of economics, which says that order in market systems is spontaneous (or emergent) in that it is the result of human action, but not the execution of any human design.
Upon this the Austrian school developed from the 19th to the early 20th century the economic calculation problem, along with the concept of dispersed knowledge, which were to fuel debates against the then-dominant Keynesian economics. This debate would notably lead economists, politicians and other parties to explore the question of computational complexity.
A pioneer in the field, and inspired by Karl Popper's and Warren Weaver's works, Nobel prize economist and philosopher Friedrich Hayek dedicated much of his work, from early to the late 20th century, to the study of complex phenomena, not constraining his work to human economies but venturing into other fields such as psychology, biology and cybernetics. Gregory Bateson played a key role in establishing the connection between anthropology and systems theory; he recognized that the interactive parts of cultures function much like ecosystems.
Topics in the complex systems study 
Challenges of managing complexity 
As projects and acquisitions become increasingly complex, companies and governments are challenged to find effective ways to manage mega-acquisitions such as the Army Future Combat Systems. Acquisitions such as the FCS rely on a web of interrelated parts which interact unpredictably. As acquisitions become more network-centric and complex, businesses will be forced to find ways to manage complexity while governments will be challenged to provide effective governance to ensure flexibility and resiliency.
Complexity Economics 
Over the last decades, within the emerging field of Complexity economics new predictive tools have been developed to explain economic growth. Such is the case with the models built by the Santa Fe Institute in 1989 and the more recent economic complexity index (ECI), introduced by the Harvard economist Ricardo Hausmann and the MIT physicist Cesar Hidalgo. Based on the ECI, Hausmann, Hidalgo and their team of the The Observatory of Economic Complexity have produced GDP forecasts for the year 2020.
Complexity and modeling 
One of Hayek's main contributions to early complexity theory is his distinction between the human capacity to predict the behaviour of simple systems and its capacity to predict the behaviour of complex systems through modeling. He believed that economics and the sciences of complex phenomena in general, which in his view included biology, psychology, and so on, could not be modeled after the sciences that deal with essentially simple phenomena like physics. Hayek would notably explain that complex phenomena, through modeling, can only allow pattern predictions, compared with the precise predictions that can be made out of non-complex phenomena.
Complexity and chaos theory 
Complexity theory is rooted in chaos theory, which in turn has its origins more than a century ago in the work of the French mathematician Henri Poincaré. Chaos is sometimes viewed as extremely complicated information, rather than as an absence of order. Chaotic systems remain deterministic, though their long-term behavior can be difficult to predict with any accuracy. With perfect knowledge of the initial conditions and of the relevant equations describing the chaotic system's behavior, one can theoretically make perfectly accurate predictions about the future of the system, though in practice this is very difficult. Ilya Prigogine argued that complexity is non-deterministic, and gives no way whatsoever to precisely predict the future (see also).
When one analyzes complex systems, sensitivity to initial conditions, for example, is not an issue as important as within the chaos theory in which it prevails. As stated by Colander, the study of complexity is the opposite of the study of chaos. Complexity is about how a huge number of extremely complicated and dynamic sets of relationships can generate some simple behavioral patterns, whereas chaotic behavior, in the sense of deterministic chaos, is the result of a relatively small number of non-linear interactions.
Therefore, the main difference between chaotic systems and complex systems is their history. Chaotic systems do not rely on their history as complex ones do. Chaotic behaviour pushes a system in equilibrium into chaotic order, which means, in other words, out of what we traditionally define as 'order'.[clarification needed] On the other hand, complex systems evolve far from equilibrium at the edge of chaos. They evolve at a critical state built up by a history of irreversible and unexpected events, which physicist Murray Gell-Mann called "an accumulation of frozen accidents." In a sense chaotic systems can be regarded as a subset of complex systems distinguished precisely by this absence of historical dependence. Many real complex systems are, in practice and over long but finite time periods, robust. However, they do possess the potential for radical qualitative change of kind whilst retaining systemic integrity. Metamorphosis serves as perhaps more than a metaphor for such transformations.
Research centers, conferences, and journals 
Institutes and research centers
- The Institute for Complex Systems and Mathematical Biology at University of Aberdeen
- The Bristol Centre for Complexity Sciences (BCCS) at University of Bristol
- International Research Center for Mathematics & Mechanics of Complex Systems (M&MoCS) at University of L'Aquila, Italy
- Center for Nonlinear Phenomena and Complex Systems at Université libre de Bruxelles, Belgium
- New England Complex Systems Institute
- Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza, Spain
- Santa Fe Institute
- Northwestern Institute on Complex Systems (NICO)
- Center for the Study of Complex Systems and Cognition at École Normale Supérieure, France
- Master's Programme in Complex Adaptive Systems at University of Gothenburg, Sweden
- Max Planck Institute for the Physics of Complex Systems, Germany
- Vermont Complex Systems Center
- Centro de Ciencias de la Complejidad, at UNAM, Mexico
- Complexity Complex at the University of Warwick, United Kingdom
- Center for Interdisciplinary Research on Complex Systems at Northeastern University
- Center for Complex Networks Research at Northeastern University
- Center for Social Dynamics & Complexity (CSDC) at Arizona State University
- Centro de Investigacion en Complejidad Social (CICS)
- Center for Complex Networks and Systems Research at Indiana University
- Southampton Institute for Complex Systems Simulation
- Center for Complexity in Business at the University of Maryland's Robert H. Smith School of Business
- Center for the Study of Complex Systems at the University of Michigan
- Center for Complexity in Health at Kent State University
- Center for Complex Systems and Brain Sciences at Florida Atlantic University
- ARC Centre for Complex Systems, Australia
- Harvard-MIT Observatory of Economic Complexity
- Center for Social Complexity at George Mason University
- York Centre for Complex Systems Analysis at University of York
- The Complexity Science Group at University of Calgary
- Plexus Institute for the study of Complex Change and Innovation
- The Center for Complex Systems Research, Univ. of Illinois at Urbana-Champaign
- The Institute of Global Dynamic Systems, Canberra, Australia
- CEiBA Complex Systems Research Center, Bogotá, Colombia
- Collective Dynamics of Complex Systems Research Group at Binghamton University, State University of New York
- CASL Institute (Complex and Adaptive Systems Laboratory) at University College Dublin, Ireland
- Advances in Complex Systems
- Complex Systems
- Interdisciplinary Description of Complex Systems
See also 
- Chapouthier, G, Mosaic structures – a working hypothesis for the complexity of living organisms, E-Logos (Electronic Journal for Philosophy), 2009, 17, http://nb.vse.cz/kfil/elogos/biocosmology/chapouthier09.pdf
- J. M. Zayed, N. Nouvel, U. Rauwald, O. A. Scherman, Chemical Complexity – supramolecular self-assembly of synthetic and biological building blocks in water, Chemical Society Reviews, 2010, 39, 2806–2816 http://pubs.rsc.org/en/Content/ArticleLanding/2010/CS/b922348g
- http://www.narberthpa.com/Bale/lsbale_dop/cybernet.htm Bale, L.S. 1995, Gregory Bateson, Cybernetics and the Social/Behavioral Sciences
- Ferguson, Adam (1767). An Essay on the History of Civil Society. London: T. Cadell. art Third, Section II, p. 205.
- Friedrich Hayek, The Results of Human Action but Not of Human Design, in New Studies in Philosophy, Politics, Economics, Chicago: University of Chicago Press, (1978), pp. 96–105.
- Bruce J. Caldwell, Popper and Hayek: Who influenced whom?, Karl Popper 2002 Centenary Congress, 2002.
- Friedrich von Hayek, The Sensory Order: An Inquiry into the Foundations of Theoretical Psychology, The University of Chicago Press, 1952.
- CSIS paper: "Organizing for a Complex World: The Way Ahead
- Reason Magazine - The Road from Serfdom
- Friedrich August von Hayek - Prize Lecture
- Hayles, N. K. (1991). Chaos Bound: Orderly Disorder in Contemporary Literature and Science. Cornell University Press, Ithaca, NY.
- Prigogine, I. (1997). The End of Certainty, The Free Press, New York.
- D. Carfì (2008). "Superpositions in Prigogine approach to irreversibility". AAPP - Physical, Mathematical and Natural Sciences 86 (1): 1–13.
- Cilliers, P. (1998). Complexity and Postmodernism: Understanding Complex Systems, Routledge, London.
- Per Bak (1996). How Nature Works: The Science of Self-Organized Criticality, Copernicus, New York, U.S.
- Colander, D. (2000). The Complexity Vision and the Teaching of Economics, E. Elgar, Northampton, Massachusetts.
- Buchanan, M.(2000). Ubiquity : Why catastrophes happen, three river press, New-York.
- Gell-Mann, M. (1995). What is Complexity? Complexity 1/1, 16-19
Further reading 
- Syed M. Mehmud (2011), A Healthcare Exchange Complexity Model
- D. Chu, R. Strand and R. Fjelland, "Theories of complexity", Complexity, 8:3, 2003
- L.A.N. Amaral and J.M. Ottino, Complex networks — augmenting the framework for the study of complex system, 2004.
- Murray Gell-Mann, Let's Call It Plectics, 1995/96.
- Nigel Goldenfeld and Leo P. Kadanoff, Simple Lessons from Complexity, 1999
- A. Gogolin, A. Nersesyan and A. Tsvelik, Theory of strongly correlated systems , Cambridge University Press, 1999.
- Kelly, K. (1995). Out of Control, Perseus Books Group.
- Graeme Donald Snooks, "A general theory of complex living systems: Exploring the demand side of dynamics", Complexity, vol. 13, no. 6, July/August 2008.
- Sorin Solomon and Eran Shir, Complexity; a science at 30, 2003.
- Preiser-Kapeller, Johannes, "Calculating Byzantium. Social Network Analysis and Complexity Sciences as tools for the exploration of medieval social dynamics". August 2010
- The Open Agent-Based Modeling Consortium
- Complexity Science Focus
- INDECS (Interdisciplinary Description of Complex Systems)
- Center for Complex Systems Research, Univ. of Illinois
- Introduction to complex systems - Short course by Shlomo Havlin
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