**Introduction**

We’ve all heard of the Butterfly Effect or maybe seen the movie called “Chaos Theory.” Unfortunately, Hollywood has led us all astray once again. The Butterfly Effect is a small part of Chaos Theory, but that’s not all it is. Chaos Theory is simply a mathematical process that helps us understand large and complex data models. Due to the sheer size of the mathematical models involved, a computer is required to simulate them. This is why Chaos Theory was not “invented” until the late 20^{th} century; we needed to invent the computer first! Examples requiring computer assisted Chaos Theory simulations include weather forecasting, migratory patterns of birds and many other areas.

**The Beginning**

It all started in the year 1960, when a man named Edward Lorenz built a weather model on his computer at the Massachusetts Institute of Technology. Given an initial set of parameters, it would compute a series of weather conditions that would never repeat. This was revolutionary in his time, and some people thought that by providing the current weather parameters, Lorenz’s program could predict the weather perfectly. Amidst all of the commotion Lorenz decided to “restart” a simulation he had run previously, but he didn’t start from the beginning as he had before, he started from a point near the middle of the simulation. The reason this was an issue was because the computer he was working on had six places of precision when working with numbers, but it only printed out three decimal places. Lorenz then used the three decimal numbers and used them as his initial starting conditions. Given the small differences between the numbers (going from six digit precision to three) he obtained vastly different results. These results were very surprising to Lorenz, but he eventually figured out his mistake. By that time, Chaos Theory was born.

**Chaos Theory Subjects**

There are a few different principles for the Chaos theory that are quite interesting. One area that I’m sure the majority of people have heard of is fractals. Fractals are essentially a pattern that repeats forever. These images can be very, very complex, like the one below.

Fractals relate to Chaos Theory because they are greatly influenced by the initial conditions. Supply a parameter that is slightly different from a previous image will result in a drastically different pattern. Another interesting thing about fractals is they appear in nature as well. If you’ve ever looked at an image where a snowflake structure has been magnified, it’s easy to see that there is a repeating pattern in the formation. Natural fractal formations also appear in the formation of tree branches and leaves, as well as some types crustacean shells.

Another principle of chaos is mixing. Mixing is the process by which two substances (molecules, balloons, anything) that start at the same location, will end up at drastically different end locations. An example of this is a water molecule in a lake. At some point in time there will be two water molecules that start near each other, but in 50 years are on opposite sides of the lake. This type of analogy can be applied to many different applications such as liquids mixing and any simulation that uses a fluid dynamics module (often called a Computational Fluid Dynamics or “CFD”).

**Conclusion**

Chaos is everywhere around us all the time, whether we like it or not. It crops up in nature and science, where it has interesting applications that cover a wide variety of topics. There is a lot of discussion on whether humankind will ever be able to fully comprehend Chaos Theory, and no one knows the answer. One thing is certain: without overcoming Chaos Theory, we will never be able to predict the weather, which is upsetting to say the least. I would like to end with a quote from Albert Einstein, who said the following:

*“As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.” –Albert Einstein*

**References**

http://fractalfoundation.org/resources/what-is-chaos-theory/

https://www.csuohio.edu/sciences/dept/physics/physicsweb/kaufman/yurkon/chaos.html