Sunday, February 05, 2006
The Butterfly and the beach
The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. The idea is that small variations in the initial conditions of a dynamical system produce large variations in the long term behavior of the system. Sensitive dependence is also found in non-dynamical systems: for example, a ball placed at the crest of a hill might roll into any of several valleys depending on slight differences in initial position.
The practical consequence of the butterfly effect is that complex systems such as the weather are difficult to predict past a certain time range - approximately a week, in the case of weather. This is because any finite model that attempts to simulate a system must necessarily truncate some information about the initial conditions—for example, when simulating the weather, one would not be able to include the wind coming from every butterfly's wings. In all practical cases, defects in the knowledge of the initial conditions and deficiencies in the model are equally important sources of error. In a chaotic system, these errors are magnified as the simulation progresses. Thus the predictions of the simulation are useless after a certain finite amount of time.
The practical consequence of the butterfly effect is that complex systems such as the weather are difficult to predict past a certain time range - approximately a week, in the case of weather. This is because any finite model that attempts to simulate a system must necessarily truncate some information about the initial conditions—for example, when simulating the weather, one would not be able to include the wind coming from every butterfly's wings. In all practical cases, defects in the knowledge of the initial conditions and deficiencies in the model are equally important sources of error. In a chaotic system, these errors are magnified as the simulation progresses. Thus the predictions of the simulation are useless after a certain finite amount of time.
