Exponential effects

The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a dynamical system may produce large variations in the long term behavior of the system. So this is sometimes presented as esoteric behavior, but can be exhibited by very simple 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 phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in a certain location. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale alterations of events. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. Of course the butterfly cannot literally cause a tornado. The kinetic energy in a tornado is enormously larger than the energy in the turbulence of a butterfly. The kinetic energy of a tornado is ultimately provided by the sun and the butterfly can only influence certain details of weather events in a chaotic manner.

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather).

"Be faithful in small things because it is in them that your strength lies." Mother Teresa