Science

The Fiftieth Anniversary of Chaos

Edward Lorenz wasn't planning to spark a scientific revolution when he published his famous paper on weather models in 1963. But the unexpected behavior of his equations opened the door to an entirely new field: chaos theory. Fifty years later, Samuel Arbesman takes a look at what this new way of scientific thinking has wrought.

Several decades ago, Edward Lorenz noticed something seemingly minor: his computational weather models yielded very different results if he changed the input variables only a tiny bit. Lorenz published a paper on this topic in 1963 and that led to the modern field of chaos.

In Physics Today, Adilson Motter and David Campbell survey the field fifty years later and what we've learned about chaos and our limits to predictability. Here's a description of the Lorenz system (and chaotic attractors in general):

A chaotic attractor is the example par excellence of a chaotic set. A chaotic set has uncountably many chaotic trajectories; on such a set, any point that lies in the neighborhood of a given point will also, with probability one, give rise to a chaotic trajectory. Yet no matter the proximity of those two points, in the region between them will lie points of infinitely many periodic orbits. In mathematical parlance, the periodic orbits constitute a countable, zero-measure, but dense set of points embedded in the chaotic set, analogous to the rational numbers embedded in the set of real numbers. Not only will trajectories that lie on the attractor behave chaotically, any point lying within the attractor’s basin of attraction will also give rise to chaotic trajectories that converge to the attractor.

Of course, chaos is found in many places far beyond weather models:

As Motter and Campbell conclude, chaos is far more than something of interest only to mathematicians or physicists:

Chaos sets itself apart from other great revolutions in the physical sciences. In contrast to, say, relativity or quantum mechanics, chaos is not a theory of any particular physical phenomenon. Rather, it is a paradigm shift of all science, which provides a collection of concepts and methods to analyze a novel behavior that can arise in a wide range of disciplines.

The rest of the paper is a wonderful overview of the nature of chaos and its many applications. For a more technical exploration, I highly recommend Nonlinear Dynamics and Chaos, by Steve Strogatz (my grad school advisor). Steve has also taught a course for The Great Courses called Chaos for a general audience.

Top image:Jared Tarbell/Flickr/CC