A process which might be described in a mathematical model that is transformed into a sequence of steps that is eventually programmed into a computer. Long division is an example of an algorithm. You could think of a recipe as an algorithm for preparing a meal.
anti-persistence
A statistical phenomenon that can be measured from data such as long term stock price fluctuations. Re-scaled range analysis can show, for example, that a data series like stock price fluctuations may be non-random and decreasing in the future if the Hurst exponent is less than 0.5.
attractor
A phase space picture of a dynamical system that depicts its movement around a region or point. Finding this region or point--called the attractor--provides scientists with proof of a system's ultimate boundaries or its final ending point.
attractor, strange
A phase space picture of a chaotic dynamical system. This picture can be measured using fractal analysis which provides scientists with a precise definition of a system's degree of complexity.
Barnsley, Michael
The inventor of the chaos game and iterated function systems (IFS). Barnsley is currently working on the application of IFS as a method of image compression for digital video.
bifurcation diagram
A graph that illustrates period doublings in population growth dynamics.
butterfly effect
The idea that very small causes can produce dramatically out-of-proportion effects. The notion that "the flap of a butterfly's wings in Brazil" might "set off a tornado in Texas" was presented in a lecture by Edward Lorenz to illustrate the impossibility of perfect weather prediction even if all known causes and effects could be measured. The butterfly effect is an illustration of sensitive dependence on initial conditions.
chaos
Certain systems, in both nature and mathematics, appear to be governed by chance, but can be shown to be deterministic through fractal analysis, phase space maps and computer models. These systems exhibit a sensitive dependence on initial conditions, so that even small variations in their starting conditions will produce wildly differing results.
chaos theory
A scientific framework which attempts to provide an understanding of mathematical models that exhibit chaotic behavior. Chaos theorists use chaos theory to search for the underlying structure of many dynamical systems in nature which seem to be complex, random, irregular and unpredictable. We are surrounded by examples of chaos, or at least things that scientists say are supposed to be chaos--a leaf fluttering to the ground, a flag flapping violently in the wind, a traffic jam, the Earth's weather and climate, and possibly the evolution of life itself.
computer model
A computer model is a list of instructions that tells a computer how to do something. A model is made essentially of four modules. The input module where some initial values are specified; the algorithm module where a mathematical process based on equations is specified and tells the computer what to do with the initial values; an output module which merely displays the results of the algorithm module as numbers or turns them into a picture; and a feedback pathway which takes output and sends it back into the input to be used by the input module as new initial values.
The computer model used to draw the Lorenz attractor is based on three equations.
x'=a(y-x)
y'=bx-y-xz
z'=xy-cz
The 3-D point, (x,y,z), is analagous to a single moment of weather and a,b,c are analagous to some of the weather's dynamic properties, i.e. the things that make weather change the way it does.
To begin drawing the Lorenz attractor, for example, it is necessary to pick some initial values for x, y and z. The shape of the attractor is determined by the parameters a, b, and c. Lorenz's choices for these were a=10, b=28 and c=8/3.
Using fairly simple and well established computer visualization techniques, a 3-D point (x,y,z) can be drawn for each new (x,y,z) generated by the model's three equations. For a sound component, three notes x, y, and z might be played.
The feedback pathway allows the computer to iterate, or repeat using the results of previous calculations over and over. Comparing these iterations gives and idea of the dynamic behavior of the mathematical model used in the program. Sometimes, the model is abstract and simple so that dynamic behavior, in general, can be examined. Many times however, the mathematical models are based on real processes, like weather, and can provide insights into the reasons why they behave unpredictably.
Because even simple models produce complex and possibly chaotic output, the output is often visualized using computer graphics and animation. The inspiring loopiness of the Lorenz attractor and the visual diversity found in the Mandelbrot set are both examples of computer visualizations of models.
computer visualization
The algorithmic rules required for transforming the geometric output of a computer model into pictures. This mapping of colors and the sequencing of distinct images (animation) on the computer screen can provide insight into the dynamic forces acting within the mathematical model that the system is based on.
convection
A basic heat transport mechanism in the atmosphere. The idea of convection is fairly simple. Part of the energy that enters the atmosphere from the sun is absorbed by the Earth's surface which heats the atmosphere from below. From above, the atmosphere is cooled when heat is radiated back into space. The lower, warmer air wants to rise upwards and the higher, cooler air wants to fall downwards. This basic mechanism creates a loop as cold air falls and hot air rises. The interactions of many such loops at different levels in the atmosphere can create turbulence and chaotic motion.
determinism
The notion that a system's present and future behavior or motion is influenced by all previous behaviors. See also butterfly effect.
dimension
An idea that helps scientists classify objects and motion. Lines and curves are examples of a single dimension which we call length; squares and circles are examples of two dimensions which we describe as area, or a length multiplied by a length; cubes and spheres are examples of three-dimensional volumes, or a length multiplied by a length multiplied by a length. These "low" dimensional objects are easy to visualize, but are limiting because scientific descriptions often require many more dimensions.
One, two, and three-dimensional measures all provide an idealized, but still useful, way of seeing and studying the world. They allow us to patch together descriptions, and make mathematical models, of the smooth and continuous portions of natural objects and phenomenon. This has been limiting as many of the things actually found in nature are unreasonably complex when examined under such simple notions of dimension.
dynamical system
A system in motion. The swing of a pendulum, boiling water, weather, the growth of populations and the interactions between atoms and molecules are all examples of dynamical systems. Some are predictable and some are not, but they are all systems whose future motions depend entirely on past movements.
equation
An expression of the relationship between mathematical objects. This expression becomes a mathematical model when the mathematical objects are identified with objects or phenomena in the real world.
feedback
See iteration/feedback
Feigenbaum, Mitchell
The mathematician who showed mathematically that the transition from simple dynamics to chaotic dynamics is universal. That is, the transition occurs (potentially) in all systems after a precise number of period doublings have occurred.
fractal
A geometrical figure consisting of a pattern that is repeated in finer and finer scales; a process or structure which is made up of similar patterns at different scales and magnifications. These processes and structures are self-similar and have a fractal dimension. They can be visualized on a computer using mathematical models and/or fractal geometry.
fractal analysis
Methods for mathematically computing the fractal dimension of fractal objects and chaotic processes. Re-scaled range analysis is an example of fractal analysis.
fractal geometry
A precise geometry used to make computer models that draw realistic looking natural objects such as mountains, trees and clouds. Using fractal analysis, such as re-scaled range analysis, the degree of complexity--or fractal dimension--of objects and processes can be precisely measured.
geometry
Traditionally defined as a branch of mathematics dealing with the function, properties and relationships of points, lines and angles, here geometry relates to distances and angles used with mathematical rules to formulate pictures. Also, mathematical rules that specify how to transform the elements of a mathematical model into a picture.
Hurst exponent
A statistical measurement provided through re-scaled range analysis that indicates the degree of non-randomness inherent in a system. A series of data like stock price fluctuations are persistent if the Hurst exponent is computer to be greater than 0.5 and anti-persistent if the Hurst exponent is less than 0.5. The fractal dimension, D, of a system with Hurst exponent, H, is 1-H.
Hurst, H.E.
A hydrologist who studied the Nile river in the 1960s. His method of statistically comparing the inflow and outflow of the Nile over many years has become one way of classifying series of data in terms of correlations between past and future events and between different time scales. Hurst's method called re-scaled range analysis applies to systems that fluctuate over long periods of time--that is, virtually anything found in nature.
iterated function system
The triangular shape generated in the chaos game raises the possibility that pre-determined rules can be used to constrain random processes to produce specific, orderly images. The triangular shape is made up of smaller and smaller copies of itself, and if it were possible to interpret real objects in terms of copies of themselves, then we might generate computer graphics of these shapes, using the same random process. For example, take the maple leaf. Although it's not made of perfect smaller copies of itself, generally it is made of similar structures at smaller and smaller scales. Making copies of the leaf, then squeezing, rotating, and shifting them allows us to specify some new rules for constraining the random process described above for the chaos game. The artificial leaf is made up of four copies of itself; one smaller version fits over the top portion, another smaller version is rotated to the left and fits the left portion and another smaller version is rotated to the right and fits the right portion. The fourth is made extremely skinny and used as the stem. To see how the leaf is made, copy these four new pieces into each other, squeezing and rotating them as needed. Continue this process until it reaches the resolution of the pixels on the computer screen.
To speed up the drawing with a computer program, you must first transform each of the leaf's pieces into a specific rule that transforms, not a square, but a single pixel chosen at random. Each rule specifies an angle of rotation; two scaling or squeeze factors: one for width and one for height; and two shift factors: one left or right, and one up or down.
The first rule says squeeze the height and width by .5 (or reduced it to half it's size), don't rotate it and move it up 40 pixels. The squeezing applied by the second rule is the same as the first, but this time rotate -45┬╛, shift left 40 pixels and up 10 pixels. The thrid rules also squeezes by the same amount, but rotates 45┬╛, shifts right 40 pixels and up 10 pixels. The fourth and final rule says squeeze by .6 and shift down 50 pixels.
Now, pick a point randomly and transform the point according to one of the rules, chosen randomly. Continue this process for 10,000 iterations, repeatedly using newly generated points as the input to other randomly chosen rules. This system of randomly chosen points and pre-defined rules is called an iterated function system. The rules are a system of mathematical functions that specify how to transform points before drawing them, and the repetitive transformation of points is called iteration.
iteration/feedback
A process by which the output of an equation is put back into the same equation over and over. Using iteration, scientists simulate the motion of dynamical systems and can then theorize how real world objects and processes become irregular and complex. Iteration refers to the algorithmic implementation of a feedback process.
long term stock prediction: global determinism vs. local randomness
Using a method of fractal analysis, called re-scaled range analysis, it is possible to examine the chaotic dynamics of long term stock market fluctuations. It shows that over long periods of time the stock market is deterministic and the average lengths of increasing or decreasing trends can be measured. It indicates that, although the stock market appears random moment-to-moment, it is actually controlled by long term memory effects created by past price fluctuations.
Lorenz attractor
A phase space picture of Edward Lorenz' chaotic dynamical model of convection. Lorenz, a meterologist, developed this model to show that weather is unpredictable.
The Lorenz model, like weather, exhibits sensitive dependence on initial conditions. For weather, this is demonstrated by any minute error in a forecasting model or in the data collected for a forecast when predictions eventually become drastically different from actual weather. For the model, this is demonstrated when the number of decimal places used for the initial input values is changed only slightly. An attractor drawn for initial (x,y,z)=(5.1234,6.4321,7.6666) will be different from initial (x,y,z)=(5.12,6.43,7.66).
Lorenz, Edward
A meteorologist at MIT who showed that weather is chaotic and ultimately unpredictable. Lorenz coined the term butterfly effect after the Ray Bradbury story, "The Sound of Thunder." His 1963 scientific paper, "Deterministic Non-periodic Flow" (J. Atmos. Sci. 20 (1963) 130-141), was one of a few that began to establish the science of chaos theory.
Mandelbrot, Benoit
A mathematician and possibly the scientist most often associated with chaos, Mandelbrot formulated the mathematics associated with fractals, and coined the term.
Mandelbrot demonstrated that real coastlines, along with all naturally occuring shapes, are not smooth like simple Euclidean lines, rectangles and triangles, but are jagged and self-similar, or fractal, and that length along the edge of a fractal shape--a coastline, for instance--is virtually infinite. From this idea arose Mandelbrot's famous question: "How long is the coastline of Britain?" For that matter, how long is any coastline?
Try this. Measure the length of your stride, then walk along a section of a pond or a lake shoreline. Then, measure the same distance with a ruler, using the foot as your unit of measurement, then cut the ruler in half, and so on for as long as you can stand it. You'll find that the distance increases with each smaller unit of measurement. Mandelbrot proved that distance depends on scale and detail. He also showed that because the coastline is so jagged, it's neither a one dimensional straight line nor a two dimensional surface. It posseses a fractional or fractal dimension, somewhere in between. With this simple idea, Mandelbrot laid out his "geometry of nature." In it, we begin find a way of expressing mathematically the irregularity of the real world.
But Mandelbrot's most ubiquitous contribution to the discipline of chaos, as well as the most demostrative example of the science's beauty and elegance, is the Mandelbrot set.
Mandelbrot set
Before computer modeling gave birth to the rendering of fractals, we could scarcely have imagined the complexity and richness that might arise from the iteration of a relatively simple mathematical equation.
This image appears when a simple set of computer instructions is applied to each of the points on a grid, such as the pixels on your computer screen. One instruction tells a computer to pick a point and then compute a new value from it. Then, other instructions continue this process, monitoring successive values for points until some appear to be only getting larger while others "orbit" a specific value. The black area, called the Mandelbrot set, shows the points whose many successive values "orbit" around a specific value. Outside the black area, colors appear when this repetitive calculating produces ever increasing values that get larger and larger without a boundary. (This image sits on a grid of 300x300 pixels, a total of 90,000 points each of which requires the generation of 2-1024 successive values.)
These abstract, but simple algorithms create an image of such complexity that scientists have yet to fully understand the mathematical significance of the image they create. However, the image is incredibly potent in it's abiltity to inspire us to make connections between chaos and the creative forces of nature.
Consider this: if a computer can create such complexity from very simple rules, maybe the natural world has a similar, underlying simplicity.
mathematical model
A system of equations that precisely links the elements that influence the behavior of a real world system. A mathematical model might eventually become a computer model that shows how the real world system behaves. Models provide scientists with a way of studying systems.
May, Robert
A theoretical biologist who demonstrated chaos in population growth dynamics.
period doublings
When a system in motion begins its transition from simple to complex behavior it does so through a pattern whereby the states of the system increase by a factor of two--they double. For example, Robert May's bifurcation diagram shows how the dynamics of population growth become more complicated when the population size fluctuates between more and more values. Initially a population may be stable at a specific number, but as the population grows over time it eventually fluctuates between 2 then 4 and 8 sizes until it becomes chaotic and fluctuates between an unpredictable number of sizes.
persistence
A statistical phenomenon that can be measured from data such as long term stock price fluctuations. Re-scaled range analysis can show, for example, that a data series like stock price fluctuations may be non-random and increasing in the future if the Hurst exponent is greater than 0.5.
phase space
An illustration of a system's motions created by plotting each of its dynamic variables on one axis of a multidimensional graph. Phase space maps were used by Henri Poincare at the turn of the century to illustrate that complex systems do not always conform predictably to the clocklike mechanisms of Newtonian physics.
Poincare, Henri
The turn-of-the-century French scientist/mathematician who showed, using phase space maps, the limitations of a Newtonian, clocklike view of the universe. This was one of the first discoveries in mathematics that planted the seeds of chaos theory.
re-scaled range analysis
A method of statistically measuring the long term factors acting on a system by comparing past data at many different time scales. The actual measure, dubbed the Hurst exponent by Benoit Mandelbrot, gives an idea of how past and future events are correlated. It also shows the average amount of time that the correlated events will last.
self-similar
A property of fractal objects and pictures of chaotic systems. When adjusting the time scale of a chaotic system or the magnification of a fractal object, similarities appear at each level. Stated simply, a coastline from 10,000 feet above looks similar to the same coastline from 10 feet above.
sensitive dependence on initial conditions
Many dynamical systems in motion--turbulent water, rising smoke, the weather--are so dynamically complex that small fluctuations can become amplified and create dramatic future effects. Because of this, exact prediction is impossible.
short term stock prediction: local determinism vs. global randomness
A two-dimensional chart shows stock market patterns over time. Hidden behind the high and lows are the forces that drive stock price fluctuations. The vast complexity of these hidden forces make the stock market a nearly impossible system to predict. It's no wonder that many economists believe that the market is completely random and can't be predicted at all. But basically, money is information. As it moves through the global economy, it leaves a trail, a pattern of information. Combined with chaos, it's possible to see that the stock market isn't entirely random. Although it might appear random over relatively long periods of time, chaos shows that some predictability is possible for short periods. That is, there might be small pockets of predictability amidst the apparent randomness.
Think of it this way. A whitewater river is a sea of vast turbulence. Examining the stock marketsΓÇÖ fluctuations reveal a similar mess of noise. Pockets of economic predictability are like a directed, and predictable, swirl in the flow of the river--for a moment, you can see just where the current is moving to next. Stock market data can be examined for the same types of recognizable patterns. Finding this ordered chaos disguised as randomness is the key to prediction.
If this seems abstract, it's because chaos isn't concerned with why patterns form, it merely looks for a pattern and tries to build a model to predict where it will end. And this is why chaos applies to so many complex systems.
Programmed into computers the mathematics of chaos theory helps in the search for valuable short term patterns that can be exploited for their financial value.
However, using the economic data that creates these kinds of charts, physicists and computer programmers look for predictable patterns using the mathematics of chaos theory. TheyΓÇÖre finding that the worldΓÇÖs economy isnΓÇÖt entirely unpredictable and may get rich off of the chaos that they discover.
turbulence
Most simply described as complex motion. Like the swirling rapids of a whitewater river, many natural systems exhibit similar kinds of complex, but structured interactions.
