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.
