@ DISCOVERY PROJECT 0.1 0.234 0.46603 0.64700 0.59382 0.62712 0.60799 0.61968 0.61276 0.61694 0.61444 0.61595 0.61505 Iteration and Chaos The iterates of a function f at a point x0 are f(xo), f(f(x0)), f(f(f(x0))), and so on. We write x1 = f(x0) The rst iterate x2 = f(f(x0)) The second iterate x3 = f(f(f(xo))) The third iterate For example, if f(x) = x2, then the iterates of f at 2 are x1 = 4, x2 = 16, x3 = 256, and so on. (Check this.) Iterates can be described graphically as in Figure 1. Start with x0 on the xaxis, move vertically to the graph of f, then horizontally to the line y = x, then vertically to the graph of f, and so on. The xcoordinates of the points on the graph of f are the iterates of f at x0. Figure 1 Iterates are important in studying the logistic function f(X) = 1060 x) which models the population of a species with limited potential for growth (such as rabbits on an island or sh in a pond). In this model the maximum population that the environment can support is 1 (that is, 100%). If we start with a fraction of that population, say 0.1 (10%), then the iterates of f at 0.1 give the population after each time interval (days, months, or years, depending on the species). The constant k depends on the rate of growth of the species being modeled; it is called the growth constant. For example, for k = 2.6 and x0 = 0.1 the iterates shown in the table to the left give the population of the species for the rst 12 time intervals. The population seems to be stabilizing around 0.615 (that is, 61.5% of maximum). In the three graphs in Figure 2, we plot the iterates of f at 0.1 for different values of the growth constant k. For k = 2.6 the population appears to stabilize at a value 0.615 of maximum, for k = 3.1 the population appears to oscillate