Tag Archives: chaos theory

Chaos Theory

butterfly

Image: RivasCAI, via Wikimedia Commons.

Chaos is the science of surprises, of the nonlinear and the unpredictable. It teaches us to expect the unexpected. Chaos Theory is the branch of mathematics that deals with complex systems whose behavior is highly sensitive to slight changes in conditions, so that small alterations can give rise to strikingly great consequences. This is what makes Chaos theory so interesting, the thought that a butterfly flapping its wings has a real connection with a cloud floating thousands of miles away.

This is called the Butterfly effect. The butterfly effect occurs under two conditions: When the system is nonlinear and when each state of the system is determined by the previous state. In other words, the output at each moment is repeatedly entered back into the system for another cycle through the mathematical functions that determine the system. Here is an example to help explain non-linearity and its connection to the Butterfly effect: “Let us take the air itself. We will assume that it is at a constant temperature and pressure and comprises atoms that collide elastically (no energy lost as heat). For how long could we predict the trajectories of the molecules (given an ideal computer)? The answer is almost no time at all, a few collisions only, a tiny fraction of a second. The system is nonlinear, as are all real systems.” (Calresco)

The two main components of chaos theory are the ideas that systems – no matter how complex they may be – rely upon an underlying order, and that very simple or small systems and events can cause very complex behaviors or events. This latter idea is known as sensitive dependence on initial conditions, a circumstance discovered by Edward Lorenz (who is generally credited as the first experimenter in the area of chaos) in the early 1960s. Complex systems are very difficult to model because it is nearly impossible to know all the initial conditions perfectly. This causes issues in predicting the final result of the complex system, one small error would be amplified dramatically.

An example of the Chaos theory in a real world settings is the stock market. This is known as a chaotic system due to feedback. As the value of a stock rises or falls, people are inclined to buy or sell that stock. This in turn further affects the price of the stock, causing a chaotic rise or fall. Another example is the double rod pendulum experiment. For this experiment you take a rod with a free flowing bend half way down, it is then dropped from any point. If you trace the bottom of the rod or have it ‘draw’ where it has been you get a graphical trajectory. Now you start the pendulum from a slightly different initial position and let it swing freely yet again. This results in a completely different graphical trajectory as the first time the experiment was conducted. The double rod pendulum is one of the simplest dynamical systems that has chaotic solutions.

The path of the double pendulum. Image: Cumi, via Wikimedia Commons.

The path of the double pendulum. Image: Cumi, via Wikimedia Commons.

If you think of every human on earth as a billiard ball, the interaction of two ‘colliding’ causes a chain of collisions that effect you. Each and every person just existing has some reaction with all the others. As a PhD math professor mentioned in his work on ‘Major open problems in chaos theory and nonlinear dynamics’. “Overall, chaos is understood but not tamed.  In fact, it is not clear whether or not it is tractable! More specifically, the mechanism of how chaotic dynamics operates is understood; how to effectively describe chaos in term of some sort of averaging (chaos engineering) is beyond reach.”(Li)

Image sources:

http://commons.wikimedia.org/wiki/File:Butterfly%27s_effect..jpg

http://commons.wikimedia.org/wiki/File:Doublependulumpath.png

Sources:

Li: http://arxiv.org/pdf/1305.2864.pdf

http://www.calresco.org/nonlin.htm

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

http://whatis.techtarget.com/definition/chaos-theory

www.econ.upenn.edu/system/files/02-02.pdf

http://www.crystalinks.com/chaos.html

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Chaos Theory: the Smallest things can have big consequences

One can tell by the name itself what chaos theory might mean. Chaos means something that is unpredictable, random and unstable. There are many known and predictable phenomena in science such as electricity, gravity or chemical reactions; however, chaos theory examines things that are not possible to control. For example, nature:  weather, earthquakes, clouds, trees, tsunamis, and tornadoes. Other than nature, there are human-related unpredictable things, such as the stock market and our brain states. Chaos theory is a field of mathematics that deals with complex systems whose behavior is highly sensitive to the slightest changes in initial conditions. For example, someone clapping their hands could change the weather, so even the smallest alterations can have big consequences.

Chaos theory emerged around the second half of the 20th century. This is because chaos theory has complex systems and these systems contain many elements that move. For this reason, computers are needed to calculate all the different possibilities.  How did chaos theory come to be? A man named Edward Lorenz, a meteorologist, created a weather model on his computer in 1960. This weather model consisted of an extensive array of complex equations to predict weather conditions. This model always gave different sequence of numbers that represented weather conditions. One day he became curious and ran his own tests to see what the outcome would be. After running a sequence, he started running the same sequence halfway through, re-entering the numbers the first sequence had given him at that point. The results were not what he was expecting; they were entirely different from his first outcomes. The second time, he entered numbers that were rounded to three instead of six digits (for example, .506 versus .506127). Since the difference between these numbers is not much, he expected the results to be only slightly varied. However, that small error gave completely different outcome. Form this he concluded that even the slightest differences in initial conditions makes prediction of past or future outcomes impossible. 

butterfly effect

Image: J.L.Westover.

There are many principles of Chaos. One of them is the butterfly effect, also described by Lorenz. It is said that even a small butterfly flapping its wings in America can create a hurricane in Japan; if the butterfly did not flap its wing at the “right” time in space then the hurricane would not have happened. Even the smallest behavior has a direct effect in the future. Another principle is unpredictability. Since it is not possible to know all the initial conditions of a complex system in adequate detail, we can’t possibly know the outcome of those. As explained above, even the smallest change in numbers can lead to a big errors in prediction; outcomes  can be completely different from what is expected.

We can never know for certain when we might have a storm or tsunami until few days before it’s about to happen. Similarly to the weather, chaos is present in our daily life. For example,  the bus you usually take was late and you decided to take another bus, and randomly you meet a person, and you both start talking, he makes an impression on you, you go on a date with him, fall in love, get married and grow old together. Now imagine that the person had a similar situation: he decided to take this bus rather than his usual bus and met you. What if he never got on that bus at the right time to meet you, and what if you had decided to wait for you usual bus? It is scary to think about how one small decision makes such a big difference in your life.

Work Cited

http://whatis.techtarget.com/definition/chaos-theory
http://fractalfoundation.org/resources/what-is-chaos-theory/
http://www.abarim-publications.com/ChaosTheoryIntroduction.html#.VFUVDfnF9Pc
http://www.crcnetbase.com.ezproxy.lib.utah.edu/doi/pdfplus/10.1201/b11408-31
http://bjps.oxfordjournals.org/content/60/1/195.full.pdf+html

Chaos Theory in action

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 20th 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.

LorenzPatterns

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.

Julia_set_(indigo)

Image: Public domain, via Wikimedia Commons.

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

http://en.wikipedia.org/wiki/Chaos_theory