# The Butterfly Effect

The other day in the car my children asked me to explain the difference between Chaos Theory and the Butterfly Effect. Questions like this are not uncommon in my family. I thought about it for a moment and realized I did not know the answer, a circumstance that I do not enjoy. So children and readers, here is the answer.

For a system to exhibit “chaos” as defined by physicists, four elements must be present. First let me list them. Then, to best of my ability, I will attempt to explain their significance. However, I must warn you that as we proceed down the list my explanations will become increasingly inept.

- The system must be deterministic.
- The behavior of the system must be very sensitive to initial conditions.
- The system must be topologically mixing.
- The system must have periodic orbits which are dense.

The first element tells us that once the initial conditions of a system are determined, the behavior of the system is also determined. Therefore, if we can reproduce the exact initial conditions, we will get the same outcome every time. This introduces you to a key point about chaotic systems, their behavior is not random.

The most dramatic aspect of chaotic systems is element number two, that small changes in initial conditions can result in big changes in the outcome. The picture at the top of the page shows a simple example of this. Let’s say that you try to place the purple ball right on the very peak of the rainbow colored shape. A very small change in the initial position of the ball can result in it rolling down an entirely different side of the hill. However, if you were successful in placing the ball in exactly the same spot each time, it would *always* roll down the same path, which is what makes this system deterministic rather than random. What gives chaotic systems the illusion of randomness is that recreating initial conditions exactly is very difficult. You may think that you are putting the purple ball in the same place each time, but actually you are not.

Once we get to element three my understanding starts to get a little murky. I don’t really understand the significance of the word “topologically” in this one. In layman’s terms, a chaotic system must be mixed. Examples include air and water molecules in a weather system, or the movement of food dye in a glass of water.

To understand element four, you need to understand the mathematics of field theory and phase space. Sadly (or perhaps happily) I don’t, so we’ll need to agree to remain entirely in the dark on this one. Any mathematicians or physicists reading this who want to help out, please use the comment space below.

The father of Chaos Theory was Edward Lorenz, a professor of meteorology and mathematics at MIT. In 1963, he was doing some weather calculations in which one of the parameters was 0.506127. The computer came up with an answer but, to be thorough, Lorenz wanted to run the program a second time to reproduce the result. Either he was in a rush or his fingers were tired so this time he rounded the parameter to 0.506 and the calculations came up with a dramatically different answer. This observation led him on a decades long quest to further explore these types of phenomena, during which he wrote a series of research papers which came to define Chaos Theory.

Just like me, Lorenz found the second element to be the easiest to explain. The first example that he used was “if a seagull flaps its wings in Tokyo it could eventually cause a hurricane in New York”. Fortunately, someone with a better ear for marketing encouraged him to use the more compelling image of a butterfly, giving birth to the Butterfly Effect.

This brings us to the answer to my children’s question. The Butterfly Effect is a subset of Chaos Theory, fulfilling elements one and two above, but not necessarily three and four. Therefore, all chaotic systems exhibit the Butterfly Effect but not all systems which exhibit the Butterfly Effect are chaotic. Not the most exciting answer in the world, but, hey, I do my best.

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