About Chaos

In the 17th and 18th centuries, Newtonian mechanics was triumphant in its explanation of the solar system. In fact, many theologians at that time pictured God as a great clockmaker, who had only to start the universe and then step back and let Newton’s laws determine the future. But does everything in the universe run like a fine clock? The nineteenth century mathematician Poincaré found an exception—the motion of three mutually interacting astronomical objects. When the three objects have comparable mass, as suggested in the drawing, the motion of each body produces a substantial change in the gravitational field experienced by the other two, thereby making a solution impossible.

Three astronomical objects moving in their mutual gravitational field, with initial velocities indicated by arrows. As each object moves, the field at the location of the others changes, making the equations that describe the motion nonlinear.

The two graphs show the development in time of a chaotic system when the initial conditions are only slightly different. Although the two curves stay together at first, they soon diverge and eventually appear to be unrelated .

This image is a phase space graph for a pendulum with friction. Each point corresponds to a state of particular position and momentum, and the succession of points around the curve correspond to the evolution of the system with time. As friction erodes the amplitude and speed of the bob, the phase space plot spirals in to the equilibrium position and zero momentum (zero speed)

The Lorenz attractor, displaying the order present in a chaotic system. The graph never crosses itself, so the system never repeats its state.
(image courtesy of Glenn Elert)

The only way to tackle this kind of problem is by a series of iterations—calculate the gravitational forces, let the objects move for a short time under these forces, then recalculate the forces based on the objects’ new positions, etc. Even without a computer, Poincaré could see that the resulting orbits were aperiodic and wildly disordered. Moreover he discovered that a small change in the initial conditions—the initial positions and velocities of the three objects—produced a huge change in the orbits. This important observation was to lie fallow for eighty years, awaiting the development of chaos theory.

Step outside and behold a chaotic system—the weather. Although there are short-term patterns, the system never repeats, and long-range forecasting remains an unsolved problem. Back in the 1960s, meteorologist Ed Lorenz build a simple computer weather model that, when given some initial values, chugged out numbers that corresponded to wind speed, precipitation, and temperature. Then this output went back in as the new input, and the model generated simple predictions over time.

Desiring to repeat a particular computer run, Lorenz restarted the computer but rounded down the initial values from six significant figures to three. Since temperatures weren’t measured to three significant figures of accuracy anyway, this approximation should have had no effect. And indeed, the output of the new run at first followed that of the old, but slowly and steadily the two diverged, as shown in the first graph, until the results were completely different and continued so, as shown in the graph. Lorenz was stunned—a change in input too small to correspond to a measurable difference had totally altered the output. Lorenz’ result exemplifies “exquisite sensitivity to initial conditions,” otherwise known as the butterfly effect—the beat of a butterfly’s wing in China could, in principle, change the weather in New York .

Lorenz presented his simulation results with a graph of what physicists call “phase space.” In this space, each axis of the graph corresponds to a system variable, such as the position and momentum of a particle, so the entire state of the system can be expressed by the point in phase space it occupies at a certain time. A pendulum with friction would have a phase space diagram like that shown in the second graph, a spiral that winds in to a point, as energy dissipates and the pendulum amplitude and speed progressively decrease. For more on the phase space of a pendulum, see the first link.

When Lorenz drew a phase space plot for his weather simulation, the results were quite different, as shown in the third graph, where the curve continues indefinitely without repeating itself and winds around and around the two points. This graph, dubbed the “strange attractor,” eventually became a potent symbol for the young field of chaos research.

Many systems show chaotic behavior. To name just a few—

• animal populations
• the red spot of Jupiter
• electrical signals in the human heart
• convection