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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.^{[1]}
The equations from which models of complex systems are developed generally derive from 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,^{[2]} economics, physics, chemistry,^{[3]} 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.^{[4]} 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.
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.^{[5]}^{[6]}
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,^{[7]} not constraining his work to human economies but venturing into other fields such as psychology,^{[8]} 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.
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.^{[9]}
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 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.^{[12]} 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 impossible to do with arbitrary accuracy. Ilya Prigogine argued^{[13]} that complexity is non-deterministic, and gives no way whatsoever to precisely predict the future (see also^{[14]}).
The emergence of complexity theory shows a domain between deterministic order and randomness which is complex.^{[15]} This is referred as the 'edge of chaos'.^{[16]}
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,^{[17]} 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.^{[15]}
Therefore, the main difference between chaotic systems and complex systems is their history.^{[18]} 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'. 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."^{[19]} 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.
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