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Unit 13

The Concepts of Chaos

13.1 Introduction

"Physicists like to think that all you have to do is say, these are the conditions, now what happens next?"

-Richard Feynman

We live in a world in which seemingly insignificant details can have a great impact. Very tiny changes in the starting conditions of a process or procedure can have substantial, sometimes even dramatic, effects on subsequent behavior and results. Some examples presented as evidence of this are purely anecdotal or even theatrical—for example, the train you miss boarding by ten seconds that ends up in a terrible crash—but others are more precise, more scientific, even mathematical. This kind of indeterminacy may seem at odds with the usual mathematical notion of a predictable world. Indeed, for centuries the prevailing view of our universe was that it "runs like clockwork," and its workings can be mathematically and even numerically predicted from a given set of starting or "initial" conditions. This predictability was possible, supposedly, because we can write equations that tell us exactly (in a perfect world) what to expect, given a set of starting circumstances. However, because we can never know anything exactly—there is always some "error" in perception or measurement—this earlier view of our world carried an implicit assumption that minor discrepancies in the measurement of those beginning circumstances are of little consequence because they should lead to only correspondingly small differences in the predicted results. As it turns out, this view is naive. The real world is one in which small differences in the initial circumstances of a sequence of events can indeed have a significant effect on the final outcome. The mathematical tools that we need to understand this sort of real-world phenomenology come from the realm of chaos theory.

Imagine that two leaves, identical in every way (size, shape, mass, texture, etc.) and attached as closely as possible to each other on the same tree branch, fall at the same time. As the leaves fall, they encounter resistance from the air, with its various eddies and small pockets of higher and lower pressure. These effects cause the two leaves to "dance" in the air as they fall. At times they are close to each other, and at other times they seem to be heading in opposite directions. They finally land in two different locations, each much farther away from the other than when they started.

How can we explain this behavior? The leaves started their descent from virtually the same location and yet ended up far apart. How could such a small difference in starting position lead to such a dramatic difference in final location?

In a linear world, this sort of behavior shouldn't happen. Had the two falling objects been apples rather than leaves, we would likely see little, perhaps no, such disparity between their initial and final separation. The density and form of the apples is such that the small shifting wind currents would have virtually no displacement effect. In linear systems such as this, outcomes are always fairly predictable if the initial conditions are known. Small differences in initial conditions, such as the spacing between the apples on the branch, result in only small differences in the eventual outcome, their spacing on the ground.

Leaves, however, are nothing like apples, and their behavior as they fall is anything but easy to explain. Their flight paths are extremely sensitive to small changes in their initial conditions. If the starting point is altered by just the tiniest amount, the path taken by a falling leaf can be entirely different. This is the hallmark of the mathematical concept of chaos.

The mathematics of chaos represents one prong of our endeavor to understand the complicated world around us. This is no small task, given the diverse complexity of our natural world—falling leaves, roiling streams, the rise and fall of species, and of course, that most unpredictable element of nature, the capricious weather. It is not hard to understand why the weather is so unpredictable; it is an extensive and vastly complicated system with many variables, all interacting in subtle ways. What's startling to realize when studying chaos theory is that even seemingly simple systems can behave in ways that are difficult to predict.

In this chapter we will learn about the mathematics of chaos and how it fits into the broader topic of nonlinear dynamics. Nonlinear dynamics can be thought of as the study of complicated things and complicated behavior. In our previous study of synchronization, we saw how individually complicated things, such as fireflies and heart cells, can behave collectively in strikingly simple ways, such as oscillating in unison. In this unit, we will see how a seemingly simple system, such as that involving a leaf falling from a tree, can exhibit extraordinarily complicated (i.e., difficult to predict) behavior. The broad field of nonlinear dynamics holds much promise for the mathematical understanding of our world. Chaos theory represents some of the first steps toward that understanding.

First, we will examine the distinction between linear and nonlinear systems. Then, we will explore the notion of predictability. From there, we will examine the fundamental trait of chaotic systems, namely, sensitive dependence on initial conditions. With these notions in hand, we will consider some examples of chaos in action.

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Next: 13.2 Linear vs. Nonlinear Systems


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