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# CHAOS THEORY

CHAOS THEORY

Chaos theory describes the behavior of certain nonlinear dynamical systems that under specific conditions exhibit dynamics that are sensitive to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, the behavior of chaotic systems appears to be random, because of an exponential growth of errors in the initial conditions. This happens even though these systems are deterministic in the sense that their future dynamics are well defined by their initial conditions, and there are no random elements involved. This behavior is known as deterministic chaos, or simply chaos.

Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, oscillating chemical reactions, fluid dynamics, mechanical and magneto-mechanical devices. Observations of chaotic behaviour 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. Everyday examples of chaotic systems include weather and climate.

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.  A related field of physics called quantum chaos theory studies non-deterministic systems that follow the laws of quantum mechanics.

As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined statistics.

Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behaviour.

Sensitivity to initial conditions is popularly known as the “butterfly effect“, so called because entitled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? 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.

For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and velocity. One might plot the position of a pendulum against its velocity. A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an orbit. Our pendulum has an infinite number of such orbits, forming a pencil of nested ellipses about the origin.

While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity.

The initial conditions of three or more bodies interacting through gravitational attraction can be arranged to produce chaotic motion.

History

In the early 1900s Henri Poincarռ, while studying the three-body problem, found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Much of the early theory was developed , under the name of ergodic theory. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

Chaos theory progressed more rapidly after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behaviour of certain experiments like that of the logistic map. The main catalyst for the development of chaos theory was the electronic computer. Electronic computers made these repeated calculations practical. One of the earliest electronic digital computers, was used to run simple weather forecasting models.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research.

Applications

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

Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of seemingly random seizures by observing initial conditions

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Butterfly effect

The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the long term behavior of the system. So this is sometimes presented as esoteric behavior, but can be exhibited by very simple systems: for example, a ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position.

The phrase refers to the idea that a butterfly’s wings might create tiny changes in the atmosphere that ultimately cause a tornado to appear (or prevent a tornado from appearing). 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.

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather).

The idea that one butterfly could have a far-reaching ripple effect on subsequent events seems first to have appeared in a 1952 short story by Ray Bradbury about time travel  .”One meteorologist remarked that if the theory were correct, one flap of a seagull’s wings could change the course of weather forever.” Later speeches and papers by Lorenz used the more poetic butterfly.

The concept of the butterfly effect is sometimes used in popular media dealing with the idea of time travel, usually inaccurately. Most time travel depictions simply fail to address butterfly effects. According to the actual theory, if history could be “changed” at all , the mere presence of the time travelers in the past would be enough to change short-term events  and would also have an unpredictable impact on the distant future, so that no one who travels into the past could ever return to the same version of reality he or she had come from and could have therefore not been able to travel in time, which would create a phenomenon known as time paradox.

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Chain reaction

A chain reaction is a sequence of reactions where a reactive product or by-product causes additional reactions to take place.

* The neutron-fission chain reaction: a neutron plus a fissionable atom causes a fission resulting in a larger number of neutrons than was consumed in the initial reaction.

* Electron avalanche process: Collisions of free electrons in a strong electric field forming “new” electrons to undergo the same process in successive cycles.

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Domino effect

The domino effect refers to a small change which will cause a similar change nearby, which then will cause another similar change, and so on in linear sequence, by analogy to a falling row of dominoes standing on end. The domino effect also relates to a chain of events.

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