Two Roads in a Yellow Wood—The Perils of Bifurcation and SDIC
If you have read Robert Frost’s Poem “The Road Not Taken” (1916), then you actually know a lot about bifurcation and sensitive dependence on initial conditions. To hit the highlights, the poem begins: "Two roads diverged in a yellow wood / and sorry I could not travel both / And be one traveler . . ." and it ends: “Two roads diverged in a wood and I / Took the one less traveled by / and that has made all the difference.” The divergence of the road is a poetic description for what is known in other circles as bifurcation:
a place of branching or forking, a choice point, where a system or person can go two different ways. Thus, bifurcation points are points of possibility, and places where chaos enters the picture.
Speaking of branches, let us take a break as Hayles (1990) describes the two branches of chaos theory: "the order out of nothing" branch and "the hidden order within chaos" or "strange attractor" branch. We have already learned about one of them, the "order out of chaos branch" of Ilya Prigogine. This branch, Hayles notes is willing to extrapolate philosophical implications and reconciles being with becoming. Compared with the strange attractor branch, the order out of nothing branch has more philosophy than results, while the strange attractor branch has more results and less philosophy. The strange attractor branch:
concentrates more on practical problems and emphasizes the ability of chaotic systems to generate new information. Almost but not quite repeating themselves, chaotic systems generate patterns of extreme complexity, in which areas of symmetry are intermixed with asymmetry down through all scales of magnification. For researchers in this branch, the important conclusion is that nature, too complex to fit into the Procrustean bed of linear dynamics, can renew itself precisely because it is rich in disorder and surprise. (Hayles, 1990, pp. 10-11)
Now, back to bifurcation. When iterations (equations where the solution of the equation feeds back into the equation on the next go round) bifurcate, that means that there are two different alternatives to an equation instead of one, and then these initial bifurcations iterate, and then bifurcate themselves. Depending upon your choices, you can end up very far from the original point and even further from where you would have ended up had you chosen the other road in the first place. Bifurcation points, as they are called, sometimes do something called period doubling, and when they do this, they quickly lead to chaos. A useful image here is the old Clairol commercial, where you tell two friends and they tell two friends and so on, except you can think of each friend as an alternative, and you can see how these different alternatives quickly beget other alternatives, all of which shows how quickly this leads to a myriad of different choices, and hence to chaos. Another image that shows bifurcations occurs at the end of the movie Love Actually (Curtis, 2003).
This period doubling phenomena causes the system to fragment into chaos, but at bifurcation points a system can also stabilize a new behavior through a series of feedback loops that couple the new change to its environment, leading to a new order (Briggs & Peat, 1989). As we can see, from iteration and bifurcation, our choices and experiences can have major consequences. Nonlinear equations are also very sensitive to where one begins, and hence the notion of SDIC.
SDIC is not some secret government agency, but an acronym for Sensitive Dependence on Initial Conditions, the idea that little things matter. A small change at the beginning of something can lead to widely divergent consequences with unpredictable results, this is also commonly known as the "butterfly effect," and we will see why, as we now continue the story of chaos theory. After, Poincaré, chaos took the road less traveled until technology caught up and changed the landscape of modern life. In the early 1960s meteorologist Edward Lorenz was trying to replicate an earlier computer run while trying to predict the weather, and went to get a cup of coffee while the computer calculated. In those days, even though Lorenz was using a supercomputer, it took a long time to run computations. When Lorenz came back the results were so different that he thought that the computer had blown a vacuum tube. What had really happened is that when Lorenz fed the numbers for the run in the second time, he fed in the numbers from the printout, and although the computer had calculated the results out to six decimal points, it only reported them back on the printout to three decimal points instead of the six. The results from this tiny difference, ten-thousandths of a point, in the starting point of these numbers led to a completely different and unpredictable result, and this extreme sensitivity to the starting point is why this idea is known as sensitive dependence on initial conditions (SDIC). SDIC explains many things, like how people who grew up in the same family can be so totally different. The term “butterfly effect” comes from the idea that something so slight of an effect as a butterfly flapping its wings in China, could over time compound and cause a change in the weather far away, for example, causing a cyclone to occur in Kansas, rather than in neighboring Oklahoma.