Chaos Theory

Studying complex and dynamic systems to reveal patterns of order (nonchaos) out of seemingly chaotic behaviors. Explanation of Chaos Theory of Lorenz ('60) and Poincaré. (ca. 1900)Contributed by: Amita Paul 
What is Chaos Theory? DescriptionThe Chaos Theory method from Lorenz and Poincaré is a technique that can be used for studying complex and dynamic systems to reveal patterns of order (nonchaos) out of seemingly chaotic behaviors. "Chaos Theory is the qualitative study of unstable aperiodic behavior
in deterministic nonlinear dynamical systems" (Kellert, 1993, p. 2). Aperiodic
behavior is observed when there is no variable, describing the state of the
system, that undergoes a regular repetition of values. Unstable aperiodic
behavior is highly complex: it never repeats and it continues to manifest
the effects of any small perturbation. Origin of Chaos Theory. HistoryIlya Prigogine, Nobel laureate, showed that complex structures could result from simpler ones. This is like order coming from chaos. Henry Adams previously described this with his quote "Chaos often breeds life, when order breeds habit". Henri Poincaré was really the "Father of Chaos [Theory]," however. The planet Neptune was discovered in 1846 and had been predicted from the observation of deviations in Uranus' orbit. King Oscar II of Norway was willing to give a prize to anyone who could prove or disprove that the solar system was stable. Poincaré offered his solution, but when a friend found an error in his calculations, the prize was taken away until he could come up with a new solution that worked. He found that there was no solution. Not even the laws of Sir Isaac Newton provided a solution to this huge problem. Poincaré had been trying to find order in a system where there was none to be found. Chaos theory was formulated during the 1960s. Significant and more practical work was done by Edward Lorenz in the 1960s. The name chaos was coined by Jim Yorke, an applied mathematician at the University of Maryland (Ruelle, 1991). Calculation of the Chaos Theory? FormulaTo apply Chaos Theory, a single measured variable x(n) = x(t0 + nt) with a starting time, t0, and a lead time, t, provides an ndimensional space, or phase space, that represents the full multivariate state space of the system; up to 4 dimensions may be required to represent the phase space for a chaotic system. Thus, over a long period of time, an observed system will develop patterns within a nonlinear time series that can be used to predict future states (Solomatine et al, 2001). Usage of Chaos Theory. ApplicationsThe principles of Chaos Theory have been successfully used to describe and explain diverse natural and artificial phenomena. Such as:
In a scenario where businesses operate in a turbulent, complex and unpredictable environment, the tenets of Chaos Theory can be extremely valuable. Application areas can include:
Steps in Chaos Theory. ProcessTo control chaos, the system or process of chaos has to be controlled. To control a system, what is needed is:
Strengths of Chaos Theory. BenefitsChaos theory has wide applicability in modern science and technology era. Communication and management may see a paradigm shift, as will several other business areas. Research and study in this area by academics can be extremely useful for the business and financial world. Limitations of Chaos Theory. DisadvantagesThe limitations of applying Chaos Theory are in due mostly from choosing the input parameters. The methods chosen to compute these parameters depend on the dynamics underlying the data and on the kind of analysis intended, which is in most cases highly complex and not always accurate. Chaos theory is not as simplistic to find an immediate and direct application in the business environment, but mapping of the business environment using the knowledge of chaos definitely is worthwhile studying. Assumptions of Chaos Theory. Conditions
Book: James Geick  ChaosMaking a new Science  Book: Ali Bulent Cambel  Applied Chaos Theory : A Paradigm for Complexity  Book: Richard Tiplady  World of Difference  Book: Garnett P. Williams  Chaos Theory Tamed 
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