30 1. Differential Equations and Their Solutions a space probe are not actually chaotic, but this same sort of control- lability has also been proved rigorously for certain chaotic systems. Experiment. Balance a broomstick vertically as best you can and let it fall. Repeat this many times, each time measuring the angle it makes with a fixed direction. You will see that the angles are ran- domly distributed around a circle, suggesting sensitive dependence on initial conditions (even though this system is not technically chaotic). Now place the broomstick on your fingertip and try to control it in a nearly upright position by making rapid slight finger motions—most people know almost instinctively how to do this. It is also instructive to note that you can make small rapid back-and-forth motions with your finger in a pre-planned direction, adding small perturbations as required to maintain the broomstick in approximate balance. (It is a fact that this actually serves to stabilize the control problem.) We hope you have asked yourself an obvious question. If the weather is too chaotic to predict, can we perhaps nevertheless control it? After all, if a tiny butterfly can really perturb things enough to cause a storm a week later, it should not be beyond the power of humans to sense the effects of this perturbation while it is still small enough to counteract. (Of course this is not an entirely new idea— people have been seeding clouds to produce rain for decades. But the real challenge is to learn enough about how large weather systems evolve to be able to guide their development effectively with available amounts of energy.) Exercise 1–21. Learn how to control the weather. Hint: It could easily take you a lifetime to complete this exercise, but if you succeed, it will have been a life well spent. 1.5.1. Further Notes on Chaos. The study of chaotic systems is a relatively new field of mathematics, and even the “correct” defi- nition of chaos is still a matter of some debate. In fact chaos should probably be thought of more as a “syndrome”—a related collection of symptoms—than as a precisely defined concept. We have con- centrated here on one particular symptom of chaotic systems, their sensitive dependence on initial conditions, but there are others that

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