1.5. Chaos—Or a Butterfly Spoils Laplace’s Dream 29 Another strange attractor, shown in Figure 1.3, appears in an ODE called the Rikitake Two-Disk Dynamo. Like the Lorenz system, the Rikitake ODE was invented to model an important real-world phenomenon, namely the Earth’s geomagnetic field. The flipping back and forth between attractor “wings” in this case corresponds to the flipping of the Earth’s North and South Magnetic Poles that has long been known from the geologic record. Figure 1.3. The Rikiatke attractor. Fortunately, even though systems that exhibit sensitive depen- dence on initial conditions do not permit long-time a priori prediction, it does not follow that such systems cannot be used to control pro- cesses that go on over long time periods. For example, when NASA sends a space-probe to a distant planet, the procedure is to look at all initial conditions and times that end up at the appropriate point on the given planet and then among these optimize for some variable (such as the transit time or payload weight). Of course they are using the prediction that with this choice of time and initial condition the probe will end up on the planet, but they realize that this prediction is only a first approximation. After lift-off, the current position and velocity of the probe is measured at intervals small enough to assure only small deviation from the previous predicted values. Then, these actual position and velocity are compared with the desired values and a “mid-course correction” is programmed that will bring the actual values back in line with the desired values. The equations governing

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