Such mathematical discoveries do seem to be borne out in the real world. Biological populations often exhibit erratic booms and busts that cannot be explained by any external cause. Long-term weather patterns defy prediction by the most powerful supercomputers. And a whole class of chemical reactions has been discovered in which the chemicals do not merely react and create a product, as they did in high school chemistry class, but oscillate back and forth between reactants and products. (Some especially nice ones cause color changes in the solution, so you can sit there and watch the stuff in the beaker go back and forth every few seconds.) The consistent story in all these discoveries is that the components of the system and their interactions themselves -- rather than any external cause -- give rise to the nonlinear behavior of the system as a whole. A rough analogy is a dozen dogs standing on a water bed. If one dog moves, he starts the bed sloshing around, which causes another dog to lose his balance and shift his weight, which sets up another wave of disturbance, until true chaos is reached.

IN the case of traffic, the German physicists -- principally Dirk Helbing and Boris Kerner, of Stuttgart -- found that given a certain combination of vehicle density and vehicle flow rate along a highway, the solution to their equations undergoes a sudden phase shift from freely moving traffic to what they call "synchronized traffic." Cars in all lanes abruptly slow down and start moving at the same speed as the cars in adjacent lanes, which makes passing impossible and can cause the whole system to jam up for hours.

In the traditional picture of traffic flow and congestion, the number of cars per minute that pass a given point on the highway at first steadily increases as the density of cars on the highway increases. (As long as everything keeps moving freely, the more cars there are on a mile of a highway, the more flow by per minute.) Eventually, however, further increases in density will cause a decrease in flow, as drivers begin braking to maintain a safe distance from the cars in front of them. A graph of flow versus density thus forms an inverted V shape. The uphill side corresponds to free flow, the downhill to congested flow. The Germans found, in effect, that under the right (or, rather, wrong) circumstances the solution to the equations can tunnel right through this hill without ever reaching the top, jumping from a state of (submaximal) free flow straight to congestion.

Such a leap from one state to another is like what happens when a chemical substance changes phase from vapor to liquid. It often happens that water in a cloud remains in the gas phase even after temperature and density have reached the point where it could condense into water droplets. Only when a speck of dust happens along, providing a surface on which condensation can take place (a "condensation nucleus"), does the transition finally occur. Helbing and Kerner basically found that free flow and synchronized flow can occur under the same conditions, and that under such "metastable" conditions a small fluctuation in traffic density can act as the speck of dust causing the shift from one to the other.