Thursday, June 24, 2004

The Butterfly Effect

Predictability: Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

The butterfly effect technically refers to the propensity of a system to be sensitive to initial conditions. It is the sensitive dependency on its initial conditions that the system, over time, becomes wholly unpredictable.

The butterfly effect is however not quite as similiar as the domino effect. For the domino effect, there is no doubt dependency of the system on the initial conditions. But a simple linear row of dominoes would merely allow one event to initiate another similar event upon each iteration. The butterfly effect however amplifies the condition upon each iteration.

The butterfly effect has been most commonly associated with the weather system as this is where the discovery of non-linear phenomenon within a complex and dynamic system began. Due to nonlinearities in the weather processes, a butterfly flapping its wings in Brazil can, in theory, produce a tornado in Texas. This strong dependence of outcomes on very slightly differing initial conditions is a distinct characteristic of the mathematical behavior known as chaos. The idea in meteorology that the flapping of a butterfly's wing will create a disturbance that in the chaotic motion of the atmosphere will become amplified eventually to change the large scale atmospheric motion, so that the long term behavior becomes impossible to forecast.

Flowing from the above argument, the butterfly effect in fact represents the essence of chaos. A complex and dynamical system is deemed to be chaotic if it
1. Has a dense collection of members with periodic orbits,
2. Is sensitive to the initial condition of the system (so that initially nearby members can evolve quickly into very different states), and
3. Is topologically transitive.
Chaotic systems exhibit irregular, unpredictable behavior. The boundary between linear and chaotic behavior is often characterized by periodic doubling in orbits, followed by quadrupling in orbits etc., although other kinds of combinations are also possible.

Animal populations are also subjected to the same phenomenon. Empirical evidence suggests that predator-prey systems too have complex dynamics making them prone to cycles. Such a system even with two simple variables such as rabbits and foxes can create a system that is really much more complex than initially thought to be. The lack of foxes may mean that the rabbit population can increase initially. But the increasing numbers of rabbits may also mean that the foxes have more food and are therefore more likely to survive and reproduce, which then in turn decreases the number of rabbits. It is possible for such systems to find a state of equilibrium, and even though species can become extinct, there is a tendency for populations to be robust. However, they can vary dramatically under certain circumstances. Real populations, of course having more than two variables, are even more complex than that of the illustration as given above.

The effects of the butterfly effect is best demonstrated by the Lorenz Attractor. The Lorenz Attractor is a graphical representation of the time variation of three variables coupled by non-linear evolution equations. You will observe that for the two separate non-linear evolution equations that are made to run simultaneously from slightly differing initial conditions, the tiny difference in the initial conditions becomes amplified by the evolution, until such time the two trajectories evolve quite separately. The amplification is exponential, the difference grows very rapidly and after a surprisingly short time the two solutions behave quite differently.

After having explained the butterfly effect from the scientific angle, it may also be appropriate to examine it from the layman's angle. There is a clever set of lyrics that is derived from an old english rhyme which originated for the purpose of encouraging children to apply logical progression to the consequences of their actions. The rhyme is often used to gently chastise a child whilst explaining the possible events that may follow a thoughtless act. But in this case, it perfectly explains the butterfly effect.

For want of a nail
For want of a nail, the shoe was lost.
For want of a shoe, the horse was lost.
For want of a horse, the rider was lost.
For want of a rider, the battle was lost.
For want of a battle, the kingdom was lost.
And all for the want of a horseshoe nail.

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