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Chaos theory

In mathematics and physics, chaos theory deals with the behavior of certain nonlinear dynamic systems that under certain conditions exhibit a phenomenon known as chaos, which is characterised by a sensitivity to initial conditions (see butterfly effect). As a result of this sensitivity, the behavior of 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.
A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3

A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3

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. Chaos theory deals with deterministic systems, and a related field of physics called quantum chaos theory studies non-deterministic systems that follow the laws of quantum mechanics.

Contents1 Description of the theory
1.1 Chaotic motion
1.2 Attractors
1.3 Strange attractors
2 History
3 Mathematical theory
3.1 Minimum complexity of a chaotic system
4 Other examples of chaotic systems
5 See also
6 References
6.1 Textbooks and technical works
6.2 Semitechnical and popular works
6.3 Popular culture

Description of the theory

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

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, for which the theory is named.

Chaotic motion

For a system to be classified as chaotic, it must have the following properties:
it must be sensitive to initial conditions
it must be topologically mixing
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. An example of such sensitivity is the so-called "butterfly effect", which suggests 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.

Topologically mixing means that 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. Since phase space is of finite size, this usually means that the phase space will become thoroughly mixed together after a fairly short time. 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.

Phase diagram for a damped driven pendulum, with double period motion
Phase diagram for a damped driven pendulum, with double period motion

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 its 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 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 its (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 Hénon 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 Poincaré-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.


The roots of chaos theory date back to about 1900, in the studies of Henri Poincaré on the problem of the motion of three objects in mutual gravitational attraction, the so-called three-body problem. Poincaré 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 LGP-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 6-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.
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.

Mathematical theory

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

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

See also


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
Deep Simplicity, by John Gribbin