For the album by Jumpsteady, see Chaos Theory (album).

For the video game, see Splinter Cell: Chaos Theory.

In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamical systems that (under certain conditions) exhibit the phenomenon known as chaos, most famously characterised by sensitivity to initial conditions (see butterfly effect). As a result of this sensitivity, the observed behavior of physical systems that exhibit chaos appears to be random, even though the model of 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, economies, 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 chaos for a discussion of the origin of the word in mythology, and other uses. When we say that chaos theory studies deterministic systems, it is necessary to mention a related field of physics called quantum chaos theory that studies non-deterministic systems following the laws of quantum mechanics.

Description of the theory

A non-linear dynamical system can, in general, exhibit one or more of the following types of behavior:

• forever at rest
• forever expanding (only for unbounded systems)
• periodic motion
• quasi-periodic motion
• chaotic motion

The type of behavior a system may exhibit depends on the initial state of the system and the values of its parameters, if any. The most difficult type of behavior to characterize and predict is chaotic motion, a non-periodic complex motion which has given name to the theory.

Chaotic motion

In order to classify the behavior of a system as chaotic, the system must exhibit the following properties:

• it must be sensitive to initial conditions
• it must be transitive
• 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 are very small. The systems behave identically only if their initial configurations were exactly the same. An example of such sensitivity is the so-called "butterfly effect", whereby the flapping of a butterfly's wings is imagined to create tiny changes in the atmosphere which over the course of time cause it to diverge from what it would have been and potentially cause something as dramatic as a tornado to occur. The butterfly flapping its wings represents a small change in the initial condition of the system which causes a chain of events leading to large-scale phenomena like tornadoes. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. Other commonly-known examples of chaotic motion are the mixing of colored dyes and airflow turbulence.

Sensitivity to initial conditions is related to the Lyapunov exponent.

Transitivity means that application of the transformation on any given Interval <math>I_1</math> stretches it until it overlaps with any other given Interval <math>I_2</math>.

Transitivity, dense periodic points, and sensitivity to initial conditions can all be extended to an arbitrary metric space. J. Banks and colleagues showed in 1992 that in the setting of a general metric space, transitivity and dense periodic points together imply sensitivity to initial conditions.

This elementary but unexpected fact prompted Bau-Sen Du, of the Institute of Mathematics, Academia Sinica, Taiwan to define a stronger version of sensitive dependence - extreme sensitive dependence - which is not a consequence of transitivity and dense periodic points. Extreme sensitive dependence means, roughly, that points close together separate and converge infinitely often, as is often the case in examples of chaotic dynamical systems.

Attractors

One way of visualizing chaotic motion, or indeed any type of motion, is to make a phase diagram of the motion. In such a diagram time is implicit and each axis represents one dimension of the state. For instance, one might plot the position of a pendulum against its velocity. A pendulum at rest will be plotted as a point and a 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.

Often phase diagrams reveal that most state trajectories wind up approaching some common limit. The system ends up doing the same motion for all initial states in a region around the motion, almost as though the system is attracted to that motion. Such attractive motion is fittingly called an attractor for the system and is very common for forced dissipative systems.

For instance, if we attach a damper to our pendulum, no matter what it's initial position and velocity it will wind up being at rest - or more correctly: it will reach rest at the limit. The trajectories on the phase diagram will all spiral in towards the middle, rather than forming sets of ovals. This point in the middle - the state when the pendulum is at rest - is called an "attractor". Attractors are often associated with dissipative systems like this, where some element (the damper) dissipates energy.

Such an attractor may be called a "point attractor". Not all attractors are points. Some are simple loops, or more complex doubled loops (for which you need more than two degrees of freedom). And some are actually fractals: the so called "strange attractors". Systems with loop attractors exhibit periodic motion. Those with more complex split loops tend to exhibit quasiperiodic motion. And systems with strange attractors tend to exhibit chaotic behavior.

At any point in on the phase diagram, the system will tend to evolve to another neighbouring state in some sort of deterministic way. If our pendulum is at a particular position and travelling with a particular velocity, we can calculate what it's (infinitesimally) "next" position and velocity will be. That is, we can treat our phase diagram as being a vector field, and use vector calculus to understand it. Attractors in our phase diagram are simply those regions with a negative divergence.

Strange attractors

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. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the Rössler Map, which experiences period-two doubling route to chaos, like the logistic map.

Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the HГ©non map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points - Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure.

The PoincarГ©-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more dimensions. However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional systems.

History

The roots of chaos theory date back to about 1900, in the studies of Henri PoincarГ© on the problem of the motion of three objects in mutual gravitational attraction, the so-called three-body problem. PoincarГ© found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. Except for Smale, who was perhaps the first pure mathematician to study nonlinear dynamics, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. 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 behavior of certain experiments like that of the logistic map. The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical. One of the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting models.

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a basic computer, a Royal McBee LPG-30, to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.

To his surprise the weather that the machine began to predict was completely different to the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 5-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.

The term chaos as used in mathematics was coined by the applied mathematician James A. Yorke.

Moore's law and the availability of cheaper computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research.

Mathematical theory

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include:

• fractal dimension of the attractor
• Lyapunov exponents
• recurrence plots
• PoincarГ© maps
• bifurcation diagrams
• Transfer operator

Minimum complexity of a chaotic system

Many simple systems can also produce chaos without relying on differential equations, such as the logistic map, which is a difference equation (recurrence relation) that describes population growth over time.

Even discrete systems, such as cellular automata, can heavily depend on initial conditions. Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30.

Other examples of chaotic systems

• Double pendulum
• Logistic map
• HГ©non map
• Lorenz model
• Smale horseshoe
• Dynamical billiards

See also

• Anosov diffeomorphism
• Bifurcation theory
• Complexity
• Dynamical system
• Fractal
  • Benoit Mandelbrot
  • Mandelbrot set
  • Julia set
• Edge of chaos
• Mitchell Feigenbaum
• Predictability

References

Textbooks and technical works

• Sprott, Julien Clinton (2003) Chaos and Time-Series Analysis, Oxford University Press. ISBN 0198508409
• Moon, Francis (1990) Chaotic and Fractal Dynamics, Springer-Verlag New York, LLC. ISBN 0471545716
• Gutzwiller, Martin (1990) Chaos in Classical and Quantum Mechanics, Springer-Verlag New York, LLC. ISBN 0387971734
• Alligood, K. T. (1997) Chaos: an introduction to dynamical systems, Springer-Verlag New York, LLC. ISBN 0387946772
• Gollub, J. P.; Baker, G. L. (1996) Chaotic dynamics, Cambridge University Press. ISBN 0521476852
• Baker, G. L. (1996) Chaos, Scattering and Statistical Mechanics, Cambridge University Press. ISBN 0521395119
• Strogatz, Steven (2000) Nonlinear Dynamics and Chaos, Perseus Publishing. ISBN 0738204536
• Kiel, L. Douglas; Elliott, Euel W. (1997) Chaos Theory in the Social Sciences, Perseus Publishing. ISBN 0472084720
• "Wave Propagation in Ray-Chaotic Enclosures: Paradigms, Oddities and Examples", Vincenzo Galdi, et. al., IEEE Antennas and Propagation Magazine, February 2005, p. 62

Semitechnical and popular works

• The Beauty of Fractals, by H.-O. Peitgen and P.H. Richter
• Chance and Chaos, by David Ruelle
• Computers, Pattern, Chaos, and Beauty, by Clifford A. Pickover
• Fractals, by Hans Lauwerier
• Fractals Everywhere, by Michael Barnsley
• Order Out of Chaos, by Ilya Prigogine and Isabelle Stengers
• Chaos and Life, by Richard J Bird
• Does God Play Dice?, by Ian Stewart
• The Science of Fractal Images, by Heinz-Otto Peitgen and Dietmar Saupe, Eds.
• Explaining Chaos, by Peter Smith
• Chaos, by James Gleick
• Complexity, by M. Mitchell Waldrop
• Chaos, Fractals and Self-organisation, by Arvind Kumar
• Chaotic Evolution and Strange Attractors, by David Ruelle
• Sync: The emerging science of spontaneous order, by Steven Strogatz
• The Essence of Chaos, by Edward Lorenz

External links

• http://www.nbi.dk/ChaosBook/
• Chaos Theory and Education
• Chaos Theory: A Brief Introduction
• Linear and Nonlinear Dynamics and Vibrations Laboratory at the University of Illinois
• The Chaos Hypertextbook. An introductory primer on chaos and fractals.
• Chaos Theory in the Social Sciences edited by L Douglas Kiel, Euel W Elliott (Google Print)ar:Щ†ШёШ±ЩЉШ© Ш§Щ„ШґЩ€Ш§Шґ

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