Chaos

Isaac Newton and the physicists of the eighteenth and nineteenth centuries who built upon his work showed that many natural phenomena could be accounted for in equations that would predict outcomes. If enough was known about the initial states of a dynamic system, then, all things being equal, the behavior of the system could be predicted with great accuracy for later periods, because small changes in initial states would result in small changes later on. For Newtonians, if a natural phenomenon seemed complex and chaotic, then it simply meant that scientists had to work harder to discover all the variables and the interconnected relationships involved in the physical behavior. Once these variables and their relationships were discovered, then the behavior of complex systems could be predicted.

But certain kinds of naturally-occurring behaviors resisted the explanations of Newtonian science. The weather is the most famous of these natural occurrences, but there are many others. The orbit of the moon around Earth is somewhat irregular, as is the orbit of the planet Pluto around the sun. Human heartbeats commonly exhibit minor irregularities, and the 24-hour human cycle of waking and sleeping is also irregular.

In 1961, Edward N. Lorenz discovered that one of the crucial assumptions of Newtonian science is unfounded. Small changes in initial states of some systems do not result in small changes later on. The contrary is sometimes true: small initial changes can result in large, completely random changes later. Lorenz's discovery is called the butterfly effect: a butterfly beating its wings in China creates small turbulences that eventually affect the weather in New York.

Lorenz, of MIT, made crucial discoveries in his research on the weather in the early 1960s. Lorenz had written a computer program to model the development of weather systems. He hoped to isolate variables that would allow him to forecast the weather. One day he introduced an extremely small change into the initial conditions of his weather prediction program: he changed one variable by one one-thousandth of a point. He found that his prediction program began to vary wildly in later stages for each tiny change in the initial state. This was the birth of the butterfly effect. Lorenz proved mathematically that long-term weather predictions based upon conditions at any one time would be impossible.

Mitchell Feigenbaum was one of several people who discovered order in chaos. He showed mathematically that many dynamic systems progress from order to chaos in a graduated series of steps known as scaling. In 1975 Feigenbaum discovered regularity even in orderly behavior so complex that it appeared to human senses as confused or chaotic. An example of this progression from order to chaos occurs if you drop pebbles in a calm pool of water. The first pebble that you drop makes a clear pattern of concentric circles. So do the second and third pebbles. But if the pool is bounded, then the waves bouncing back from the edge start overlapping and interfering A head-on collision between two dipolar vortices entering a stratified fluid environment from the right and left sides of the picture. The original vortices have exchanged some of their substance to form two new mixed dipoles which are moving at roughly right angles to the original direction of travel (toward the top and bottom of the photo). Dipolar vortices are relevant to turbulence in large-scale geophysical systems like Earth's atmosphere or oceans. Turbulence within a fluid is an example of a chaotic system. Photograph by G. Van Heijst and J. Flor. Photo Researchers, Inc. Reproduced by permission. with the waves created by the new pebbles that you drop in. Soon the clear concentric rings of waves created by dropping the first pebbles are replaced by a confusion of overlapping waves.

Feigenbaum and others located the order in chaos: apparently chaotic activities occur around some point, called an attractor because the activities seem attracted to it. Figure 1 illustrates an attractor operating in threedimensional space. Even though none of the curving lines exactly fall one upon the other, each roughly circular set of curves to the left and right of the vertical line seems attracted to an orbit around the center of the set of circles. None of the curved lines in Figure 1 are perfectly regular, but there is a clear, visual structure to their disorder, which illustrates the structure of a simple chaotic system.

James Yorke applied the term "chaos" to non-linear dynamic systems in the early 1970s. But before Yorke gave non-linear dynamical systems their famous name, other scientists had been describing the phenomena now associated with chaos.