Why does a butterfly flapping its wings in Brazil could set off a tornado in Texas?
The Butterfly Effect has always been an intriguing concept and is often used in more philosophical context than in mathematical. But, unfortunately, the origin of this concept and its history is unknown to many. Through this article, I will give a short qualitative introduction to Chaos Theory and its origins to people with little or no mathematics background.
Chaos is generally considered as disorder or randomness in the system. However, it is more truly a stochastic (we’ll understand stochastic in a while) behaviour observed in deterministic systems. Now there are majorly two types of models: Deterministic Model and Stochastic Model. Deterministic is when you can predict the changes in the output whenever there is a change in the input variables. There is a set of equations describing the relation between input and output. One such example of a Deterministic Model is Newton’s Laws (if you ignore Drag Forces). Newton’s Laws simply imply that whatever happens in the future is determined by what happens now. You can accurately predict the orbit of planets around the sun, the path of projectiles all thanks to the Deterministic Model of Science.
Stochastic, on the other hand, deals with randomness and you can never predict the output. It’s like tossing an unbiased coin. There is an equal probability of both heads and tails.
Now there are systems like weather, which have multiple variables like wind speed, humidity, pressure, rotation of the Earth etc. are sort of deterministic. However, in such a system, a tiny fluctuation in the input variable results in a large variation in the output.This variation is not random, hence, it falls in the realm of a Deterministic model. A little error in the measurement of any variable can create a huge fluctuation in the result. We call it Deterministic Chaos or Chaos Theory.
Weather was how Chaos Theory was first discovered. Edward Lorenz is considered as the Father of Chaos Theory. He was one of the firsts to combine Mathematics and Metrology and created a model using a set of 12 differential equations to forecast weather. He used variables like atmospheric pressure, wind speed, etc. in his equations. In 1961, Lorenz was attempting to re-run a simulation of his data. While entering the data, he used a shortcut by rounding off the data to the nearest thousands. The number entered was 0.506 instead of 0.506127. He was expecting a duplicate result, however the output was completely different from the previous result. It is due to how chaotic systems function. They magnify the result even on the tiny changes in the input of that system (next time be careful when you casually round off numbers).
It is, thus, impossible to predict an outcome of a system at a far away point in time. That is why most of the weather reports turn out to be wrong. This sensitivity to the initial variables makes it difficult to forecast the weather past a certain point in time. This is what is referred to as The Butterfly Effect. The weather prediction models are inaccurate because it was impossible to know the exact starting conditions. In his later interviews, Lorenz used this butterfly analogy to explain the theory. “A butterfly flapping its wings in Brazil could set off a tornado in Texas.”
In other words, the flapping of the wings would result in a very, very tiny change in the atmospheric pressure (not observable) and this change will magnify in the output. Hence, the flapping of wings (tiny change in atmospheric pressure) at a place results in a setting off a tornado at somewhere else.
These indiscernible changes having huge implications in the Chaotic systems led Lorenz to conclude that it was impossible to predict the weather. There was another factor in play which is called Lorenz Attractor. When the weather simulation was run over and over with minuscule variations, the patterns that were visualised were not overlapping onto itself and it was circling the empty areas of space.
It resembled a butterfly with its 2 wings. This is what differentiates a chaotic system from a random system. Weather forecast beyond a certain time is useless. Next time, if the weather forecast turns out to be wrong, blame the Chaos Theory. The other classic examples of Chaotic Systems are Stock Market Pattern, Double Pendulum, and Beating of the heart.
Now going back to the beginning of my article where I mentioned how the Butterfly Effect is used quite as an analogy in philosophy, sports, and life. One field which I think the Chaos Theory describes perfectly is FC Barcelona’s modern gameplay which is often called as Tiki Taka.
After watching Barcelona in recent years, their current Tiki Taka can be very well defined as a chaotic system. Tiki Taka gameplay is successful when the opponent let Barcelona have the ball. No change in the input! The same expected behaviour in every match. As we know, chaotic systems deliver prodigious output on a tiny change in the input, this is something Barcelona managers need to consider while developing their gameplay strategy. The modern football game is now about quick transition and more second balls. All the opponent has to do is press close. Tiki Taka breaks down when pressed with aggression. No wonder they lost the second tie so brutally against Roma and then with Liverpool. Organized Chaos is the enemy of Tiki Taka. Being a Barcelona fan, I hate to bring this analogy but for the sake of my conscience, it had to come up. This is a classic chaotic system.
If I managed to spark an interest in the subject, refer Nonlinear Dynamics and Chaos, a book by Steven Strogatz, for further reading. It is one of the finest literatures on the subject.
