Chaos Theory: Chaos Within?

Chaos theory is a field of study in mathematics, physics, and philosophy studying the behavior of dynamical systems that are highly sensitive to initial conditions.

This sensitivity is popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future dynamics are fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable.
This behavior is known as deterministic chaos, or simply chaos.
Chaotic behavior can be observed in many natural systems, such as the weather.

Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.

Chaotic behavior can be observed in many natural systems, such as the weather. Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.■

One of the most successful applications of chaos theory has been in ecology, where dynamical systems such as the Ricker model have been used to show how population
growth under density dependence can lead to chaotic dynamics.Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions.
A related field of physics called quantum chaos theory investigates the relationship between chaos and quantum mechanics.

The correspondence principle states that classical mechanics is a special case of quantum mechanics, the classical limit. If quantum mechanics does not demonstrate a sensitivity to initial conditions, it is unclear how exponential sensitivity to initial conditions can arise in practice in classical chaos. Recently, another field, called relativistic chaos,has emerged to describe systems that follow the laws of general relativity.The initial conditions of three or more bodies interacting through gravitational attraction
can be arranged to produce chaotic motion.■

Chaos theory is applied in many scientific disciplines: mathematics, biology, computer science, economics,engineering,
finance,philosophy, physics, politics, population dynamics, psychology, and robotics.

Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions,fluid dynamics, and mechanical and magneto-mechanical devices

Observations of chaotic behavior in nature include the dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, the dynamics of the action potentials in
neurons, and molecular vibrations.

There is some controversy over the existence of chaotic dynamics in plate tectonics and in economics.■