Can the flap of butterflies affect hurricane season?
April 24th, 2008, 2:00 am · Post a Comment · posted by grobbins
Welcome to the latest guest column from Daniele Struppa, chancellor at Chapman University in Orange. Struppa also is a mathematician.
Can the flap of a butterfly in Australia cause a hurricane in the Atlantic Ocean?
The recent news of the passing of mathematician and meteorologist Edward Lorenz, 90, at his home in Cambridge, Mass., on April 16 gives us an opportunity to revisit one of the most famous (and probably poorly understood) physics phenomenon, the so-called Butterfly Effect.
Some readers may be familiar with a 1952 short story of Ray Bradbury, “A Sound of Thunder,” where the inadvertent killing of a butterfly in a prehistoric world ends up changing the future of the world as we know it (the spelling of English itself is one of the many long-range consequences of that single death). Our younger readers who are less familiar with printed work, and more at ease with recent TV shows, may be aware of at least one movie based on this concept: the unimpressive 2004 flick “The Butterfly Effect,” starring Ashton Kutcher.
More commonly, the term Butterfly Effect usually refers to the idea that the flapping of a butterfly’s wings creates small changes in the atmosphere, which could then amplify and ultimately cause a hurricane.
But is this effect true, and can it be predicted in any significant way?
Back in the early sixties, Edward Lorenz was working on a computer model for weather forecasting. In one simulation, he decided to make a very small change in the initial conditions of the model as a way to simplify the calculations. Much to his surprise, and against what was the consensus, the weather forecast changed completely! Lorenz then realized that even if we have a very accurate mathematical model of a phenomenon (in this case the evolution of the weather), it is possible that in some situations very small changes in the initial conditions will lead to wildly different results.
This discovery was so significant that Lorenz published it in a 1963 paper. Even though Lorenz did not use the catchy story of the butterfly, this phenomenon eventually became known as the butterfly effect, possibly because of the title that was given to one of his lectures at the American Association for the Advancement of Science in 1972.
While this effect may appear counterintuitive, we are all quite familiar with such a phenomenon. Suppose, for example, that we are crossing a street from a traffic light to another. Clearly, small variations in our starting point will lead to only small variations in our arrival point. There is no reason for us to carefully plot step by step our street crossing. No matter what we do (within reason) we will reach the other side of the street. However, suppose now that we are crossing a river walking on a tightrope, or on a balance beam. Then, we all know that even a very small change in the way in which we place our first step can have catastrophic results, and we may end up in the water below!
What is the difference between the two physical systems we have described? Simply stated, the first system (the crossing of a road) is a stable system, in the sense that all the directions are essentially equivalent, and because of this the “equations” which describe our motion are not susceptible to changes if we change slightly the initial conditions. On the other hand, the system which describes the crossing of a river on a tightrope is an unstable system. This means that the outcome changes wildly if I change, even minimally, my first (or second) step.
So, the phenomenon is neither new, nor surprising. What was surprising and extremely important in the work of Lorenz was the understanding that weather phenomena are, by their own nature, unstable, and the equations which describe them are such that (like the walk on a tightrope) small changes will lead to huge variations in the outcome. What was further surprising in the work of Lorenz, was the realization that such an unstable situation may occur even when dealing with equations which mathematicians would normally regard as rather simple and innocuous looking.
The work of Lorenz was one of the starting points for what is now known as chaos theory. Despite a name which seems to indicate randomness and confusion (the Greek word chaos actually means “primal emptiness” and not disorder or confusion as we commonly believe), chaotic systems are all but random. Rather they exhibit a behavior which mathematicians call “deterministic”; by this word we mean that the evolution of the system may appear to be inexplicable, but is in fact precisely determined by its initial conditions.
So, in the case of weather forecast, the system is not governed by chance, and if we knew its initial conditions (temperature, barometric pressure, wind speed, etc.), we could predict, in principle, its evolution with complete precision. However, in practice, we do not have access to the exact initial conditions, and the small and unavoidable experimental errors are such that our forecast is hopelessly flawed.
Scientists have now found that chaos (often much more complex than the one originally described by Lorenz) underlies the behavior of a variety of systems ranging from electrical circuits, to population growth, from weather and climate, to the stock market. Mathematicians, on the other hand, have developed tools that allow them to “manage” the behavior of chaotic systems, and in fact to use such behavior to better understand the nature of the phenomenon at hand. Applications of chaos theory now reach in a variety of different disciplines, somewhat in a paradoxical contrast to the nature of the early work of Lorenz. What began as an unsettling discovery, has now (almost fifty years later) become one of the most active and important areas of applied mathematics.
