In 1961, Edward Lorenz discovered chaos in the clockwork universe.
Lorenz was running a computer simulation of the atmosphere to help forecast the weather. He wanted to rerun just part of one sequence, so instead of starting at the beginning, he started the run in the middle. In order to start in the middle, he used the output of the program at its midpoint as a new starting point, expecting to get the same result in half the time.
Unexpectedly, even though he was using a computer, following strict deterministic rules, the second run started with the same values as before but produced a completely different result. It was as if uncertainty and variability had some how crept into the orderly, deterministic world of his computer program.
As it turned out, the numbers used from the middle of the run were not quite the same as the ones used when the program was at that same point the first time because of rounding or truncation errors. The resulting theory, Chaos Theory, described how for certain kinds of computer programs, small changes in initial conditions could result in large changes later on. These systems change over time where each condition leads to the next. This dependence on initial conditions has been immortalized as “the butterfly effect†where a small change in initial conditions- the wind from a butterfly’s wings in China, can have a large effect later on- rain in New York.
This sensitivity to the exact values of parameters in the present makes it very hard to know future values in the future. As its been formalized mathematically, chaos theory applies to “dynamical system†which simply is a system that changes over time according to some rule. The system starts in some initial state at the beginning. For our purposes, think of it as now, time zero. Rules are applied and the system changes to its new state- wind is blowing, temperature is changing, etc based on rules applied to the initial state of the atmosphere. The rules are applied to the new state to produce the next state, etc.
Chaos may not have been the best word to describe this principle though. To me it suggests complete unpredictability. Most real or mathematically interesting dynamical systems don’t blow up like that into complete unpredictability. Using the weather, for example, even if the butterfly or small differences in ocean surface temperature makes it impossible to know whether the temperature in TImes Square in New York will be 34 degrees or 37 degrees on February 7th, either one is a likely value to be found in the system at that time in that place. Measuring a temperature of 95 degrees F in New York in February is impossible or nearly so.
Dynamical systems like the weather often show recurrent behavior, returning to similar but non-identical states over and over as the rules are applied. Following the values over time describes a path that wanders over values, returning after some time to the same neighborhood. Not exactly the same place because it started in a slightly different place than the last time around, but in the same neighborhood. Just unpredictably in the same neighborhood.
This returns us to the distinction between knowing the future and predicting it. The future state of chaotic system can be known because small changes in initial conditions result in large changes in result. But those large changes recur in a predictable range of values. A chaotic system can be predicted even though its future state can’t be known. When it comes to the TImes Square temperature, climate data tells us what range the chaotic values move within from season’s cycle to season’s cycle. In drug development, the chaotic system of taking a pill every day and a measuring drug levels in the blood allows prediction as the range of likely values, but because initial conditions change and cause large, unpredictable effects one can’t know in advance whether today’s measure will be high or low. Its almost never the average, it varies around the average.
It’s important to see how important prediction is for making decisions when the future is unknown. Because the uncertain future is orderly we actually know a lot about it; we don’t know it in all of its particulars. We must make decisions knowing what range of possibilities the future can assume. Chaos Theory suggests that this kind of uncertainty is in the very nature of the world because of the behavior of dynamical systems, any rules dictate how a system changes over time.
