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Stability of a fixed point Let us imagine a concave floor dipping down gradually toward the center of a gym. Then, no matter where you drop a soccer ball, it eventually comes to rest at the center hollow. This is illustrated in figure 4 for 3/4 = 0.75000…, a ball being released at the height of 7, say, ft. bounces once to 5 ft. high and then comes to rest thereafter. Hence, the center hollow of floor acts as a fixed point introduced in Lesson 1. That is, once trapped in the center hollow a ball remains there forever. Also, it is a stable fixed point in that a ball rolls back down to the center hollow after you give a gentle kick to it. Stability is an important thing in everyday living. In fact, it is reassuring to know that we can steer a car without losing control (except on ice, of course) and the Earth will not spin out of the solar system.
Figure 4. Concave floor
Strangely, there is no word for the opposite of being stable, and hence we say it is not stable or unstable. Let us now suppose the opposite case of a floor being raised up at the center of gym, as shown in figure 5. A ball dropped on such a convex floor would roll off to the edges. In particular, a ball dropped at the center apex of floor rolls off to either the right or left edge in an unpredictable fashion. This is because uncontrollable factors such as, a minute deflection from the straight downward path, a slight deviation in the floor curvature, a sudden puff of air movement, etc., can send the ball rolling off to right or left, as if though by a coin tossing. Indeterminacy is the hallmark of instability, and this is how chaos creeps in and we are often surprised by instability (Lesson 10).
Figure 5. Convex floor
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