Chaos - James Gleick [24]
For a simple system like a pendulum, the phase space might just be a rectangle: the pendulum’s angle at a given instant would determine the east-west position of a point and the pendulum’s speed would determine the north-south position. For a pendulum swinging regularly back and forth, the trajectory through phase space would be a loop, around and around as the system lived through the same sequence of positions over and over again.
Smale, instead of looking at any one trajectory, concentrated on the behavior of the entire space as the system changed—as more driving energy was added, for example. His intuition leapt from the physical essence of the system to a new kind of geometrical essence. His tools were topological transformations of shapes in phase space—transformations like stretching and squeezing. Sometimes these transformations had clear physical meaning. Dissipation in a system, the loss of energy to friction, meant that the system’s shape in phase space would contract like a balloon losing air—finally shrinking to a point at the moment the system comes to a complete halt. To represent the full complexity of the van der Pol oscillator, he realized that the phase space would have to suffer a complex new kind of combination of transformations. He quickly turned his idea about visualizing global behavior into a new kind of model. His innovation—an enduring image of chaos in the years that followed—was a structure that became known as the horseshoe.
MAKING PORTRAITS IN PHASE SPACE. Traditional time series (above) and trajectories in phase space (below) are two ways of displaying the same data and gaining a picture of a system’s long-term behavior. The first system (left) converges on a steady state—a point in phase space. The second repeats itself periodically, forming a cyclical orbit. The third repeats itself in a more complex waltz rhythm, a cycle with “period three.” The fourth is chaotic.
To make a simple version of Smale’s horseshoe, you take a rectangle and squeeze it top and bottom into a horizontal bar. Take one end of the bar and fold it and stretch it around the other, making a C-shape, like a horseshoe. Then imagine the horseshoe embedded in a new rectangle and repeat the same transformation, shrinking and folding and stretching.
The process mimics the work of a mechanical taffy-maker, with rotating arms that stretch the taffy, double it up, stretch it again, and so on until the taffy’s surface has become very long, very thin, and intricately self-embedded. Smale put his horseshoe through an assortment of topological paces, and, the mathematics aside, the horseshoe provided a neat visual analogue of the sensitive dependence on initial conditions that Lorenz would discover in the atmosphere a few years later. Pick two nearby points in the original space, and you cannot guess where they will end up. They will be driven arbitrarily far apart by all the folding and stretching. Afterward, two points that happen to lie nearby will have begun arbitrarily far apart.
SMALE’S HORSESHOE. This topological transformation provided a basis for understanding the chaotic properties of dynamical systems. The basics are simple: A space is stretched in one direction, squeezed in another, and then folded. When the process is repeated, it produces a kind of structured mixing familiar to anyone who has rolled many-layered pastry dough. A pair of points that end up close together may have begun far apart.
Originally, Smale had hoped to explain all dynamical systems in terms of stretching and squeezing—with no folding, at least no folding that