What is sensitive dependence on initial conditions?
Consider a boulder precariously perched on the top of an ideal hill. The slightest push will cause the boulder to roll down one side of the hill or the other: the subsequent behavior depends sensitively on the direction of the push–and the push can be arbitrarily small. Of course, it is of great importance to you which direction the boulder will go if you are standing at the bottom of the hill on one side or the other! Sensitive dependence is the equivalent behavior for every initial condition–every point in the phase space is effectively perched on the top of a hill. More precisely a set S exhibits sensitive dependence if there is an r such that for any epsilon > 0 and for each x in S, there is a y such that |x – y| < epsilon, and |x_n - y_n| > r for some n > 0. Then there is a fixed distance r (say 1), such that no matter how precisely one specifies an initial state there are nearby states that eventually get a distance r away. Note: sensitive dependence does not require exponentia
