Strange Attractors

Writer: 
October 1, 2009
A child on a swing

Mischa Photo Ltd.

Wherever chaos is found in the world around us—weather patterns, dripping faucets, swirling smoke—something more than mere randomness is at work. Despite their unpredictability, chaotic systems also display surprising regularities, patterns encapsulated by one of chaos' most intriguing beasts—the strange attractor. An attractor, in a physical system, is the collection of all behaviors that system is capable of settling into. Take, for example, a child swinging alone on a playground. With no one to push the swing, friction will eventually bring it to rest. This "steady state"—with the swing at a dead hang—is the attractor of the swinging-child system. Whatever the starting point, direction, and speed of the swing, the child will always wind up in the same place, see Figure 1.

Now consider the case of an adult pushing the child in a regular rhythm. No matter how the swing starts, so long as the adult keeps pushing, the child will eventually end up swinging back and forth in sync with the push. In this second case, swinging in time with the push is the attractor of the system, see Figure 2 .

When mathematicians study a physical system, they often do so by plotting its attractors. As the variables of the system change over time, they draw a line connecting their values at each instant. This line then represents the history of the system's motion, and the pattern it eventually follows is the attractor. In the case of the hanging swing, the attractor is a single point, representing no motion. From any starting position or speed, the line that describes the system is inevitably "attracted" to it. In the case of the adult pushing the child, the attractor is a closed loop, representing repeating, periodic motion, see Figure 3.

Diagrams explaining attractors
 
 

Chaotic systems, however, are a different story. Because chaotic systems display aperiodic behavior—motions that never exactly repeat—the lines representing their evolution never overlap or intersect. Yet, instead of moving randomly, the lines describing the system trace out roughly the same pattern over and over—in the case of one of the most famous chaotic systems, a double loop reminiscent of an opera mask or butterfly's wings, see Figure 4. No matter how the system starts, its behavior eventually comes to trace out this pattern, which the mathematicians David Ruelle and Floris Takens dubbed a "strange attractor."

Mathematically, the presence of a strange attractor offers a way to distinguish chaos from mere randomness, providing a way to identify new chaotic systems in nature. Today, computergenerated images of strange attractors remain the most widely recognized symbols of the study of chaos.