Complexity_ A Guided Tour - Melanie Mitchell [16]
When R is approximately 3.569946, the values of x no longer settle into an oscillation; rather, they become chaotic. Here’s what this means. Let’s call the series of values x0, x1, x2, and so on the trajectory of x. At values of R that yield chaos, two trajectories starting from very similar values of x0, rather than converging to the same fixed point or oscillation, will instead progressively diverge from each other. At R = 3.569946 this divergence occurs very slowly, but we can see a more dramatic sensitive dependence on x0 if we set R = 4.0. First I set x0 = 0.2 and iterate the logistic map to obtain a trajectory. Then I restarted with a new x0, increased slightly by putting a 1 in the tenth decimal place, x0 = 0.2000000001, and iterated the map again to obtain a second trajectory. In figure 2.10 the first trajectory is the dark curve with black circles, and the second trajectory is the light line with open circles.
FIGURE 2.10. Two trajectories of the logistic map for R = 4.0: x0 = 0.2 and x0 = 0.2000000001.
The two trajectories start off very close to one another (so close that the first, solid-line trajectory blocks our view of the second, dashed-line trajectory), but after 30 or so iterations they start to diverge significantly, and soon after there is no correlation between them. This is what is meant by “sensitive dependence on initial conditions.”
So far we have seen three different classes of final behavior (attractors): fixed-point, periodic, and chaotic. (Chaotic attractors are also sometimes called “strange attractors.”) Type of attractor is one way in which dynamical systems theory characterizes the behavior of a system.
Let’s pause a minute to consider how remarkable the chaotic behavior really is. The logistic map is an extremely simple equation and is completely deterministic: every xt maps onto one and only one value of xt+1. And yet the chaotic trajectories obtained from this map, at certain values of R, look very random—enough so that the logistic map has been used as a basis for generating pseudo-random numbers on a computer. Thus apparent randomness can arise from very simple deterministic systems.
Moreover, for the values of R that produce chaos, if there is any uncertainty in the initial condition x0, there exists a time beyond which the future value cannot be predicted. This was demonstrated above with R = 4. If we don’t know the value of the tenth and higher decimal places of x0—a quite likely limitation for many experimental observations—then by t = 30 or so the value of xt is unpredictable. For any value of R that yields chaos, uncertainty in any decimal place of x0, however far out in the decimal expansion, will result in unpredictability at some value of t.
Robert May, the mathematical biologist, summed up these rather surprising properties, echoing Poincaré:
The fact that the simple and deterministic equation (1) [i.e., the logistic map] can possess dynamical trajectories which look like some sort of random noise has disturbing practical implications. It means, for example, that apparently erratic fluctuations in the census data for an animal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors: they may simply derive from a rigidly deterministic population growth relationship such as equation (1)…. Alternatively, it may be observed that in the chaotic regime arbitrarily close initial conditions can lead to trajectories which, after