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. That’s nothing but 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.