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The Butterfly Effect

It has been said that “little things are merely the causes of great things” and that the smallest actions can result in tremendous consequences. Therefore, some people argue that there are no such things as “little” or “insignificant” in the world that we live in because even the tiniest change or alteration in our current circumstance has a resounding impact in our future.

It’s a fascinating philosophical perspective, but did you know that there is an existing mathematical concept that somehow operates partly under the same basic principles? What I am referring to is the mathematical phenomenon in chaos theory referred to as “sensitive dependence on initial conditions.” While the official term may be a little too wordy, you may be familiar with this concept with its more popular and catchier term – “the butterfly effect.”

If you “google” search this mathematical term, you’ll find plenty of materials online that are related to films, books, music, and articles about philosophy and religion, all of which somehow allude to an oversimplified version of what the “butterfly effect” is. And if you’ve watched, the 2004 movie “The Butterfly Effect,” which starred Ashton Kutcher, you probably got to see a crude illustration of how the smallest changes in a person’s life can result in large consequences in the grand scheme of things over time.

Of course, the brilliant and scientific minds in the field of mathematics and physics will probably tell you that there is so much more to this mathematical idea than that, but it has been a challenge to adequately transmit what it means to the non-scientific population of our modern society. And so, for this video, let’s try to bridge – even just by a small fraction – the gap between mathematicians and the general population in their understanding of what the butterfly effect is.


Now, before we delve deep into the definition of the butterfly effect, let us briefly talk about what chaos theory is in the field of mathematics. Chaos theory is a branch of study in mathematics that is directed at analyzing the behavior of dynamical systems which are substantially sensitive to initial conditions. Essentially, it deals with things which are nonlinear and unpredictable, and therefore difficult to anticipate and control. It is concerned with deterministic systems whose behavior, in theory, can initially be predicted but seems to become random over time.

American mathematician and meteorologist Edward Norton Lorenz, who went down in history as one of the pioneers of chaos theory, described chaos as “when the present determines the future, but the approximate present does not approximately determine the future.”

Chaotic behavior is a relevant consideration in the study of different natural systems, including weather and climate. The theory is also applied in various disciplines including, but not limited to, environmental science, biology, computer science, engineering, and economics.


The humble beginnings of the butterfly effect as one of the mathematical concepts of chaos theory can be traced back more than 50 years ago when Edward Lorenz was a professor of meteorology at MIT and was crunching numbers through a computer program that allowed the simulation of weather patterns. On this history-making day, Lorenz was redoing the simulation he ran earlier that day, only this time, one variable was rounded off to the thousandth decimal place. He discovered that this minute alteration significantly changed the predictive pattern that his computer program had produced, resulting in an entirely different weather scenario.

It is this surprising simulated outcome that led Lorenz to reach a stunning revelation regarding how nature operates – that tiny changes can lead to big consequences. And so, in 1963, Lorenz published his findings in his paper titled “Deterministic Nonperiodic Flow,” which challenged Isaac Newton’s classical idea of a “clockwork universe.” Newton suggested the predictability of every aspect of the universe as it is a perfect system controlled by the laws of physics. For Lorenz, there is unpredictability even in a deterministic sequence. His work soon gained ground in the 1970s and the 1980s and became a founding principle of chaos theory.

The mathematical concept of the butterfly effect is encapsulated in this proverbial question: “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” This was the title of a talk in 1972 conducted by Lorenz during which he explained that a butterfly flapping its wings could result in tiny alterations in the earth’s atmosphere and will consequently prevent, delay or accelerate the formation of a hurricane or tornado in a distant place.

This does not mean that the butterfly itself possesses the power to create a tornado and direct where it goes. Instead, the theory implies that the flapping of the butterfly’s wings is one of the initial conditions of the Earth’s weather system. This means that in one set of conditions, the flapping wings in Brazil could over time led to the occurrence of a tornado in Texas. And so, had the creature not flapped its wings, the tornado would never have formed. However, it is important to note that it is just as likely that a set of initial conditions which do not feature the butterfly flapping its wings could be the one that gives rise to a tornado.

Here lies the problem posed by the butterfly effect concerning mathematical prediction. In the real world, initial conditions for a particular system can never be identified with 100% accuracy. For example, all the variables that influence weather conditions, like temperature and wind speed, cannot be known completely. Instead of an exact prediction, we are forced to provide an ensemble of forecasts which have been mathematically calculated under known but also imperfect initial conditions.

Hence, Lorenz’s butterfly effect boils down to unpredictability. In a non-linear system, it is impossible to make predictions about the future unless we possess perfect knowledge of the initial conditions. And should we try to, even the slightest change or error in the established initial conditions could lead to an entirely different destiny.


This is probably why popular culture is fond of the idea of the butterfly effect. Many of us believe that everything happens for a reason, and we hope that we can identify those reasons, no matter how minute they may be. And this where pop culture’s take on the butterfly effect gets it wrong. Nowadays, the original meaning of the butterfly effect has been lost along the way since it became a favored subject matter of mass culture. Instead of stressing the unpredictability of nonlinear dynamic systems, a large number of people from the non-scientific population gained a misguided understanding of the mathematical concept with their supposition and expectation that the smallest reasons or a chain of seemingly insignificant events can potentially alter history and form new destinies.

A popular example often raised to explain the butterfly effect is the speculation over the spark that ignited the First World War. Some people say that it all boils down to a driver making a wrong turn while driving the car of the Archduke of Austria, Franz Ferdinand. This mistake led to the assassination of the Archduke and his wife, which was followed by Austria-Hungary’s declaration of war against Serbia. What came after that was the Germans declaring war on Russia, then France, Belgium, and the UK went to war to fight Germany. By pop culture’s understanding of the butterfly effect, World War 1 never would have happened the Archduke not been assassinated, and he would not have died if his driver did not make a wrong turn.

Some of us would like to think that a trivial course of action was the one that triggered a series of events that resulted in a catastrophic consequence which resulted in the death and suffering of many people. However, what Lorenz’s mathematical theory is telling us is that it would have been impossible for us to accurately predict with absolute certainty whether World War 1 was going to happen when it did. The archduke’s driver making a wrong turn may have led to the start of the First World War, but then again, it would not have made a difference anyway. If the driver didn’t make that mistake, perhaps the war would have just been delayed, or maybe we would have been doomed to experience an even more terrible global armed conflict than the one that took place in real history. We would have just made several forecasts of what could have happened, but we would have never gotten every little thing right.

Before he died, Lorenz himself revealed that he was unsure of the proper answer to his question of whether a butterfly’s flapping wings can indeed cause a tornado. To him, the value of the question he raised decades ago lies in the bigger point it evokes – that nature’s web of cause and effect are often just too convoluted to unravel as it is highly sensitive to tiny changes.

And while we cannot accurately predict future events, Lorenz’s butterfly effect concept has inspired significant advancements in various scientific fields as scientists are now less inclined to underestimate the intrinsic complexity of the world’s multitude of systems, from the atmosphere to the stock market. For the last few decades, modern science has evolved from the classical emphasis on stability, permanence, and predictability to the new-age recognition that our everyday lives are filled with instability, sensitivity, and unpredictability. And so, the butterfly effect is more than just a metaphor or a mathematical concept; it is now a symbol of modern science’s new and improved state of mind.



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