# Bus Schedules in Chaos

Mathematics might explain why your bus never comes when you expect it.

The other day, at a bus stop in San Francisco, an electronic sign counted down the minutes until the 33-Ashbury would arrive. At last, the ticker concluded that the bus was "Arriving." Yet, despite whatever the sign might have said, the bus didn't show up for another five minutes. Why, you ask? Maybe, according to new research, it's because of chaos.

Chaos theory, that is. To mathematicians and complex systems scientists, chaos doesn't mean confusion or disorder; chaos is what happens when very slight changes to a system lead to dramatically different outcomes. Chaotic systems are characteristically hard to predict, even when the laws of physics or mathematics tell you exactly what will happen next. A good example is the double pendulum—one pendulum hanging off another one—which might wiggle gently back and forth one moment and spin wildly the next. (Here's a nice video explanation, courtesy of MIT.)

Buses aren't toys or table-top physics demonstrations, of course, but they do share some features of chaos. Sometimes you miss one bus and wait 20 minutes for the next, and sometimes they come one right after another. Never, it seems, do they show up exactly when you expect them to.

### The key is the minimum time it takes the bus to travel one block relative to the time it takes a red light to turn green.

But what could make bus times so unpredictable? To investigate, a team of researchers led by Jorge Villalobos, dean of natural sciences and mathematics at the University of Ibagué, in Colombia, developed a model in which a single bus travels down a street stopping only for red lights or to pick up passengers mid-block. Traffic lights were spaced at regular intervals and timed so they all turned red or green at once.

Despite the model's simplicity, fomenting chaos was easy. The key, the researchers found, is the minimum time it takes the bus to travel one block relative to the time it takes a red light to turn green. When those times are equal, the bus is synchronized with the lights, and it moves along smoothly.

Something different happens when the red lasts a little longer, or if the bus moves a little faster. That, Villalobos writes in an email, throws the bus slightly out of sync with the traffic lights. "When you are getting near the light ... you have to start braking, but before you stop the light goes to green and the [bus] starts accelerating again," Villalobos explains. Do that "stop and go" over and over, he explains, and chaos ensues (i.e. your bus is late).

"Practically, it means that one day you commute for a given time, the next day longer, another one shorter, and you can't explain why; same bus, same starting time, same traffic lights, same weather, no accidents [but] different commuting times," Villalobos says.

While real-world traffic is clearly more complicated than the researchers have assumed, they've already arrived at two intriguing, though as yet unpublished, results: Aggressive drivers cause more chaos, and chaos makes for poor fuel economy. Something to ponder the next time your driver slams on the brakes only to floor it when the light says go.

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