There are essentially two types of systems in the physical universe. Predictable systems are those whose behavior can be completely determined from mathematical equations. An example is an oscillating pendulum. Its time period, position of bob at any future time instant, etc. can be determined from the equations that govern oscillatory motion. Such systems are also called Newtonian systems. On the other hand, complex or unpredictable systems are those whose behavior cannot be completely determined from mathematical equations, although approximations can be made. A good example is the weather. We cannot predict what the weather will be more than a few days into the future. This is because such systems are extremely sensitive to initial conditions. This can be best illustrated from the story of how Chaos Theory was first discovered.

In 1960, a meteorologist by the name of Edward Lorenz was developing a mathematical system for weather prediction. He had developed a model using 12 equations which could predict the weather up to a few days ahead. Once, he wished to run a particular weather sequence again. To save time, he started in the middle of the sequence. He supplied values off a print-out of the sequence and let it run. When he came back after some time, he saw that the new sequence was vastly different from the previously computed one. On closer inspection, he found that to save time, he had entered the values to only the first three decimal places (0.506), instead of all the six decimal places that were given on the print-out (0.506127). The diagram below shows both sequences together. Notice how the two sequences diverge vastly towards the end, even though at the initial point, the difference between them was just 0.000127!

This made Lorenz realise that the mathematical model he had designed was extremely sensitive to initial conditions. Indeed, if such a tiny difference could result in a completely different sequence of effects, how could we predict the weather beyond a few days when present-day measurement systems were not so accurate as to allow reliable measurement of necessary parameters (temperature, atmospheric pressure, etc) up to more than a few decimal places? This condition of sensitive dependence on initial conditions came to be known as the Butterfly Effect, the analogy being that a butterfly flapping its wings in one part of the world causes a shift in conditions enough to markedly affect the future weather in another part of the world. The theory governing the behavior of such systems is called Chaos Theory.

Such sensitive systems are extremely difficult to analyze accurately with mathematical models because it is simply impossible to practically achieve the level of accuracy of parameter measurement required for correct analysis. The logical question is: how do we then analyze such systems mathematically? The answer is: although it is rather difficult to determine the exact state of such systems at any future instant, it has been observed that these systems typically exhibit a recurring pattern of states in their behavior, and hence, it is possible to determine the range of values within which the state of the system may lie at a future instant by analyzing previous variations in the state of the system. Such recurrence must however not be confused with periodicity, because although they exhibit recurring patterns or cycles, the individual patterns are never the same, an essential condition for a system to be periodic. If they were, the system would be predictable and we would be rid of a major headache!

To illustrate this concept statistically, if we make a 3-D plot of the values a system takes over a long interval, we observe an underlying regularity in its behavior. Although it never really repeats its behavior exactly, and is therefore not periodic, it does lie within a fixed range of values which can be determined from statistical analysis of previous states. For a practical example, let us revert to the weather prediction case discussed earlier. We roughly expect the weather to be hot in summer and cold in winter, and although we can’t really determine the exact values of temperature for next summer, it is possible to determine the range of values from meteorological data of previous years. This is Chaos Theory at work!

Chaos Theory has found numerous applications, ranging from weather forecasting to analyzing stock market trends to studying human populations and even creating music! Even washing machines that claim to use "fuzzy logic" to wash clothes better actually do so by using basic concepts of Chaos Theory to generate random pulsator movements.