Explaining traffic jams February 7, 2008Posted by dorigo in mathematics, news, science.
I just finished browsing a paper by Gabor Orosz and Gabor Stepan, researchers respectively in nonlinear mathematics and applied mechanics. The pdf file had rest on the virtual desktop of my laptop computer for a while now, begging to be read like a hundred more, but advantaged by having not been thrown to the darkness of my “papers to read” folder with all the others. And today, with some time to spend before the arrival of the next train to Venice, I just ventured to read it.
After my quick read I am left with mixed feelings. The paper is not the kind of science that fits George Bernard Shaw’s definition, which I learned from Jeff a week ago: “Science is always wrong! It never solves a problem without creating ten more“. In fact, it does answer the question of how traffic jams are created from a uniform flow. The problem in this case is that the answer was already rather well known. Nonetheless, just thinking at the elegant math which is the ultimate cause of your anguish at the wheel when stuck on a highway makes it easier to accept the situation, and this is enough justification for the article. But the study is indeed some breakthrough in modeling traffic jams.
Orosz and Stepan consider an idealized highway such as the one pictured on the right: vehicles are the points at coordinates along a circumference, all moving in the same direction. They then analyze the nonlinearities that arise in a model of traffic flow in their toy highway when one introduces a realistic time delay in the response of drivers to the detection of an impact threat with the car preceding them. They find that the time delay is crucial in allowing to model, with quite complicated formulas, the onset of backward-traveling “stop-and-go” waves, which interrupt the unstable solution of a well-behaved uniform flow of vehicles.
Apparently, the duality between uniform flow and “stop-and-go” waves has a name: it is an instance of a Hopf bifurcation. Now, since I had never heard of Hopf bifurcations before (or maybe I have, and have forgotten about them – oblivion is the privilege of a cultured man), I am not the best person to explain it here.
So you can read about it on wikipedia if you can not stand your own ignorance (I have accepted mine long ago). If you have lost your mouse and cannot click above, here is a quote:
In bifurcation theory a Hopf or Andronov-Hopf bifurcation is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane.
Everything is clear now, huh ? Well, the math is really not for everybody, not even in the simplest case. And it turns out that the time delay introduced by Orosz and Stepan changes the description of the system from one with ordinary differential equations in a finite-dimensional dynamical space to one modeled by delay differential equations and infinite-dimensional phase spaces. Hugh.
In any case, however complex the main body of the paper is, its conclusions are quite readable. Basically, the model of Orosz and Stepan demonstrates the onset of backward-traveling waves of traffic jams, and shows how a highway is basically a bistable system, with the linear flow easily affected by large enough “perturbations” -.such as a truck changing lane – which cause the onset of stop-and-go waves. All things we knew, but the formulas in the model do allow some planning: just a little decrease in the speed of cars approaching a backward-moving wave could significantly dampen it. Something we knew qualitatively, but we can now compute. A step in the right direction, towards the fulfilment of my highway dream.
I dream of highways where you enter with your car, and then leave the wheel and the gas pedal and read a book. An electronic wireless system controls the speed of your car and its steering, and you get to destination in the smallest possible time available given the number of vehicles on the road. This is not science fiction: we have owned the technology to do this since maybe ten years ago. Just imagine the amount of time saved to human beings, the decrease in pollution, and in the amount of stress… I know these systems are being studied, and I am rooting for those guys.