Catching the flutter of a falling leaf.

With the approach of autumn in the Northern Hemisphere, falling leaves are in the air -- and in the scientific literature.

The falling motions of a leaf or sheet of paper are very complex, say Yoshihiro Tanabe and Kunihiko Kaneko of the University of Tokyo. Sometimes, the leaf or paper may drift randomly to the left or right as it falls. At other times, it may tumble erratically while maintaining a downward course. Its motion can also be quite regular.

To investigate whether paper's irregular motions can be attributed to chaos, Tanabe and Kaneko developed a simplified mathematical model of a sheet of paper failing through the air. Their analysis of the model demonstrates that tiny changes in the amount of friction acting on falling paper can induce unpredictable, erratic behavior, suggesting that its motion is chaotic.

The most direct way to take into account the complicated interactions between air and a sheet of paper is to use the Navier-Stokes equations, which describe the motions of a fluid. But such an approach, even with approximations, would require huge expenditures of computer time.

Instead, Tanabe and Kaneko focused on a simple model with only a few variables to capture the motion's essential features. They simplified the motion so that it takes place in two dimensions and assumed the paper has a length and mass but no width or thickness.

In addition, only three forces act on the paper: lift, friction, and gravity. The friction is divided into two components, one acting parallel and the other perpendicular to the direction of fall. The researchers set the perpendicular component to be always larger than the parallel component.

They used this crude approximation to study the effect on the falling paper's motion of increasing the amount of friction in the perpendicular direction. They clearly distinguished five falling patterns.

When the friction force is weak, the paper drifts to one side, steadily rotating or flipping over as it falls. For slightly higher values, this rotation turns into erratic tumbling.

As the friction increases, the paper begins to flutter, swaying chaotically from side to side during its downward course. At still higher values, the swaying motion becomes regular (see illustration).

Finally, the paper's sideways movements decrease to zero, and it falls straight down. It's like the fall of a needle in honey, the researchers note.

"We have succeeded in constructing a simple model reproducing a variety of falling patterns observed in daily life," Tanabe and Kaneko conclude in the Sept. 5 Physical Review Letters.

It may be possible to test this model experimentally -- perhaps by dropping a thin, flat object in fluids of different densities -- to confirm the chaos of fluttering fall.