The Cairo Geniza is an accumulation of almost 200,000 Jewish manuscripts that were found in the genizah of the Ben Ezra synagogue (built 882) of Fustat, Egypt (now Old Cairo), the Basatin cemetery east of Old Cairo, and a number of old documents that were bought in Cairo in the later 19th century. These documents were written from about 870 to as late as 1880 AD and have now been archived in various American and European libraries. The Taylor-Schechter collection in the University of Cambridge runs to 140,000 manuscripts, a further 40,000 manuscripts are at the Jewish Theological Seminary of America.
Khan el-Khalili is an ancient bazaar, or marketplace adjacent to the Al-Hussein Mosque. It dates back to 1385, when Amir Jarkas el-Khalili built a large caravanserai, or khan. (A caravanserai is a hotel for traders, and usually the focal point for any surrounding area.) This original carvanserai building was demolished by Sultan al-Ghuri, who rebuilt it as a new commercial complex in the early 16th century, forming the basis for the network of souqs existing today. Many medieval elements remain today, including the ornate Mamluk-style gateways. Today, the Khan el-Khalili is a major tourist attraction and popular stop for tour groups.
Topological mixing (or topological transitivity) means that the system evolves over time so that any given region or open set of its phase space eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.
Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.
In chemistry, predicting gas solubility is essential to manufacturing polymers, but models using particle swarm optimization (PSO) tend to converge to the wrong points. An improved version of PSO has been created by introducing chaos, which keeps the simulations from getting stuck. In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets. Four of the five moons of Pluto rotate chaotically. In quantum physics and electrical engineering, the study of large arrays of Josephson junctions benefitted greatly from chaos theory. Closer to home, coal mines have always been dangerous places where frequent natural gas leaks cause many deaths. Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled, can be predicted fairly accurately.
Chaos theory can be applied outside of the natural sciences. By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, better suggestions can be made to people struggling with career decisions. Modern organizations are increasingly seen as open complex adaptive systems with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. The chaos metaphorused in verbal theoriesgrounded on mathematical models and psychological aspects of human behavior provides helpful insights to describing the complexity of small work groups, that go beyond the metaphor itself.
It is possible that economic models can also be improved through an application of chaos theory, but predicting the health of an economic system and what factors influence it most is an extremely complex task. Economic and financial systems are fundamentally different from those in the classical natural sciences since the former are inherently stochastic in nature, as they result from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data. The empirical literature that tests for chaos in economics and finance presents very mixed results, in part due to confusion between specific tests for chaos and more general tests for non-linear relationships.
Traffic forecasting also benefits from applications of chaos theory. Better predictions of when traffic will occur lets measures be taken to disperse it before it would have occurred. Combining chaos theory principles with a few other methods has led to a more accurate short-term prediction model (see the plot of the BML traffic model at right).
Chaos theory can be applied in psychology. For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in Wilfred Bion's theory is a basic assumption, the group dynamics is the result of the individual dynamics of the members: each individual reproduces the group dynamics in a different scale, and the chaotic behavior of the group is reflected in each member.
Chaos theory has been applied to environmental water cycle data (aka hydrological data), such as rainfall and streamflow. These studies have yielded controversial results, because the methods for detecting a chaotic signature are often relatively subjective. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics.