Chaos theory

Source: http://en.wikipedia.org/wiki/Chaos_theory

In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. Among the characteristics of chaotic systems, described below, is a sensitivity to initial conditions (popularly referred to as the butterfly effect? ). As a result of this sensitivity, the behavior of systems that exhibit chaos appears to be random, even though the system is deterministic in the sense that it is well defined and contains no random parameters. Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, and population growth.

Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. (See the article on mythological chaos for a discussion of the origin of the word in mythology, and other uses.) A related field of physics called quantum chaos? theory studies non-deterministic systems that follow the laws of quantum mechanics.

Chaotic dynamics

• 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 two points in such a system may move in vastly different trajectories in their phase space even if the difference in their initial configurations is very small. The systems behave identically only if their initial configurations are exactly the same.

Sensitivity to initial conditions is popularly known as the "butterfly effect", suggesting that the flapping of a butterfly's wings might create tiny changes in the atmosphere, which could over time cause a tornado to occur. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

Sensitivity to initial conditions is often confused with chaos in popular accounts, but in itself is not particularly interesting. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behavior, as all points except 0 tend off to infinity.

Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system.

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• See also: Cellular Automaton