Is life predictable? Isn't it like weather forecast? Yes, it is partially predictable in the short term. However, the accuracy of the long term forecast is way off and not many people take it seriously. Even in the short term forecast, it is only partially predictable. Sometime a slight cause exerts a big effect in the weather pattern. Our life is very much like it. In the late 20th century, there is hardly an earthshaking thought created except the Theory of Chaos. It is a mathematical model of studying a system related to its stability. The Chaos states that the butterfly effect is the sensitive dependency on initial conditions in which a small change at one place in a deterministic nonlinear system can result in large differences in a later state.
In the course of our life, the successful rate of the short term trip is very high comparing to the long term one. It can be understood by using an input/output model with a deterministic box A, input A(i) yield output A(o). For illustrated purpose, a person plans with input A(i) = 5. He thinks it is 5 exact. But in reality, the input value may be 5.00 with accuracy to two significant digits. In the real world, a point is like a real number. The person thinks his input is 5.00 but actually it may be 5.003 or 5.00243 in the real situation. He can't see the difference in input as he only sees to the two digit accuracy in decimal place. The point here is that even the person can only control the input to certain decimal places, any slight difference in the nth decimal places may cause a big change in the output for a nonlinear system. For example the person may control his input to 5.000000.... But the sudden death of his son in a remote area may be the 100th decimal place of his input. Since life most of the time is a nonlinear system, a small perturbation may cause a big change of behavior. In this case, it is really look like a butterfly effect as something occurs thousand miles away but the impact is almost instant at the Internet speed. The result is that the person has to terminate the trip he planned and rush to his sun's funeral. The outcome A(o) is totally out of prediction from his input but can be explained with the Theory of Chaos. In real life, we simply can not control the input to the accuracy of reality and the nth decimal place very often causes a big effect to the outcome. It can also be rephrased as: When the present determines the future, but the approximate present does not approximately determine the future.
Since history is the story of mankind (a group of people), its behavior can also be unpredictable. In the field of Electrical Engineering, there is one discipline called Control System. It studies the stability of a system and how to control it. A Russian Aleksandr Lyapunov (1857-1918) figured out some way to determine whether the system is stable without actually solving the detailed states of the system. Lyapunov's study and impact were significant, and it is interesting to know a number of different mathematical concepts and engineering terms bear his name. Here are some examples, Lyapunov Equation, Lyapunov Function, Lyapunov Exponent, Lyapunov Stability, Lyapunov Vector, Lyapunov Time etc. Maximal Lyapunov Exponent (MLE) determines a notion of predictability for a dynamical system. A positive MLE is taken as an indication that the system is chaotic.
Chaos theory also concerns some deterministic systems whose behavior can in principle be predicted. Chaotic systems are sometime predictable for a while and then appear to become random. The amount of time for which the behavior of a chaotic system can be effectively predicted depends on three factors:
(1) how much uncertainty we are willing to tolerate in the forecast
(2) how accurately we are able to measure its current state
(3) a time scale depending on the dynamics of the system, called the Lyapunov time.
Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a couple of days; the solar system, 50 million years. In chaotic systems the uncertainty in a forecast increases exponentially with elapsed time. Hence doubling the forecast time squares the proportional uncertainty in the forecast. This means that in practice a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears to be random. Perhaps the significance of Lyapunov's work is that someday we may prove the human history to be chaotic without getting into a lot of details in analyzing it.
