Chaos Details Page Chaos Details

Attractors	Loosely speaking, an attractor is what "naturally" happens. More precisely, an attractor is a state of the system with this stability property: if the system starts near the state, it will move closer (get attracted) to the state. For the glass sculpture to the left, standing on its base is an attractor. If the glass is tilted slightly, it will wobble some but quickly return to the position shown. There are two important notes to add. First, this system has other attractors (e.g., lying on its side). Second, there are states which are equilibria but unstable (e.g., balancing on top of its neck). An attractor doesn't have to be an equilibrium state (see the discussion below), but it does have to be stable.
Strange Attractors	The discovery in the 1960's of "strange attractors" launched chaos theory as an important research field. The toy shuttle to the left has two component motions: the shuttle can spin around on its central rod, and the outside circular frame can rotate around or rock back and forth. The outside frame is similar to a pendulum: because of gravity, one attractor is resting with its heavy side down. However, a magnet in the base of the toy adds energy to the system, giving us another attractor: it can rock back and forth at a constant rate. For many years, mathematicians thought that these were the only attractors possible: stationary or periodic. The shuttle has a different type of attractor: a chaotic (often called strange) attractor. On a strange attractor, the motion is non-constant and aperiodic. That is, the path of the shuttle does not repeat and is essentially unpredictable (this is why the toy is fun). Thus, the "natural" state of a system, even a simple system, can appear to be random and chaotic.
Sensitive Dependence	Imagine two flies buzzing around a room. The rules of fly-buzzing might well produce a strange attractor. That is, the flies' paths might be aperiodic and bounded. First, notice that aperiodicity forces the paths to be intricate: if you have to fly around for eternity without ever passing through the same point twice, you will have to follow a very complicated path. This path will be a fractal, as discussed below. Also, the flies are likely to come very close to each other every now and then, but they will then veer away from each other. This illustrates sensitive dependence to initial conditions: two paths initially very close together will quickly diverge. The remarkable fact is that this occurs on all strange attractors. The most famous example of this is the weather. The physical laws governing the weather produce a strange attractor of weather patterns that never quite repeat. Edward Lorenz discovered the weather's sensitive dependence to initial conditions and gave the first description of what is now commonly called the butterfly effect. The air stirred by a butterfly flapping its wings in Brazil could be the start of a chain of events resulting in a tornado in Texas.
Bifurcations	One of the goals of chaos research is to determine the conditions under which different types of attractors occur. All systems have parameters that affect the performance of the system. For example, the motion of a pendulum is affected by its length and the amount of friction at its pivot. If there is friction, then the only attractor is the pendulum's vertical resting position. If there is no friction, then another attractor (periodic motion) exists. The value of a parameter at which the set of attractors changes is called a bifurcation point. Some cardiologists now think of fibrillation as a bifurcation from healthy heart dynamics to unhealthy dynamics; the defibrillator provides an electrical pulse to shock the heart back to a healthy rhythm. An important parameter in this process is the ratio of the frequencies of signals sent out by the two pacemaker nodes of the heart.
Fractals	Human beings have remarkable visual processing abilities. One of the most significant aspects of the computer revolution is the increased computing power to turn jumbles of messy data into graphics that we humans can analyze. Chaos theory required this power to come of age as a scientific discipline. The Mandelbrot set shown to the left is one example. This image illustrates the behavior of the mathematical system x2+c for the complex parameter c. For example, if c=0.2, you iterate the function x2+.2. Starting at x=0, compute 02+.2=.2, then .22+.2=.24, then .242+.2=.2576 and so on. Do several more iterations and you will see the iterates reach about .276 and stop changing. By contrast, try c=0.5. This time, iterate the function x2+.5. Starting at x=0, compute 02+.5=.5, then .52+.5=.75, then .752+.5=1.0625 and so on. The next few iterates are 1.6, 3.1, 10.4 and 109.5. These iterates become unbounded (blow up). The Mandelbrot set is the set of all parameters c such that the iterates of x2+c remain bounded. This set, which is colored blue to the left, has an infinitely detailed boundary and contains multitudes of miniature near-perfect copies of itself. The structure of the set, with bulbs branching off of bulbs, gives precise information about the nature (equilibrium, periodic or chaotic) of the system's finite attractor. This is an amazing set!
Nature	As much fun as the Mandelbrot set and other fractals are mathematically, the field of chaos didn't take off until IBM researcher Benoit Mandelbrot, meteorologist Edward Lorenz and others began finding chaos and fractals in the real world (see James Gleick's excellent book Chaos for a description of several of the chaos pioneers). Fractals (infinitely detailed geometric shapes which often contain miniature copies of themselves) can be found in the branching processes of trees, human lungs, human vascular system, coastlines, clouds and almost any object formed by numerous iterations. Fractals are being used to provide highly efficient data compression algorithms, realistic computer graphics, reliable encryption codes, and to identify patterns in complicated systems (e.g., the stock market) that had previously resisted mathematical analysis.