The Butterfly Effect

Can a butterfly flapping its wings effect a tornado halfway across the world?  Can it greatly alter future events and the evolution of time?

The butterfly effect is the concept that the evolution of a system may be highly sensitive to perturbations in the initial data.  It is one of the main conditions for chaos.  The term was coined by Edward Lorenz, one of the pioneers of chaos theory, in a paper titled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? The idea is that a small change in the initial conditions, such as a butterfly’s flap of its wings may cause large scale changes over time leading to a tornado in some other part of the world.

The idea is featured prominently in time travel movies such as appropriately named The Butterfly Effect, Run Lola Run, or the Back to the Future movies, where some small change in the current timeline results in drastically different results at some later time.  It’s interesting to wonder whether in fact, one’s life would be drastically different if some or even one event in one’s life were slightly different.  In Jurassic Park, the chaos theorist Ian Malcom, illustrates the concept by a drop of water rolling down one’s hand when placed near the knuckle.  Even a slight change in where the drop of water is placed results in the drop moving in different directions and larger distances apart than the initial distance.

Systems such as weather exhibit this behavior, and thus make it difficult to accurately predict weather beyond just a few days.

Mathematically speaking, the conditioning of a system is a quantitative measure of it sensitivity–how slight perturbations in the initial conditions propagate.  The condition number is defined as , and can be understood to be an amplification factor that relates the error or change in a system’s evaluation relative to error or change in a system’s input.  If the relative change in the output of the system is large compared to its input (large condition number), then the system is ill-conditioned or sensitive.  If the relative change in the output of a system is small or commensurate with the input (small condition number), than the system is well-conditioned or insensitive.

The conditioning of a system also depends on the input data.  For example, in the water drop example given above, the system is ill-conditioned near the knuckle, but anywhere else, the water drop will flow in the same direction regardless of small perturbations, and thus, the system is well-conditioned.