Chaos Theory: A Primer Into Sensitivity to Initial Conditions
With all the talk about this crazy weather around the world (yeah, i'm sure global warming doesn't have anything to do with it), it primes the way for me to describe to you the amazing concepts contained within chaos theory. It is in part because of chaos theory that it is so very difficult to accurately predict this weather. Here, I will introduce some of the basic concepts of chaos theory and the butterfly effect.
While describing these systems as "chaotic" makes it seem as if the outcome of a system could never be determined, they can, although it is really hard to do so. Despite our efforts to predict the weather and other chaotic phenomena, these systems are actually deterministic in nature. That is, the outcome (for example, the end result at the end of a specified amount of time, say T=60) is already laid out from the moment the system either started or began to be measured (T=0) based upon the starting conditions as present between physical forces and matter. But these systems are also dynamical. It is because of this that these systems appear random or "chaotic".
Chaos theory can be summed up by reading the title of this article, as one of the most important criteria of chaotic systems is their "sensitivty to initial conditions". This concept is logical yet fascinating. It simply states that changing the starting conditions, however so slightly, can produce dramatic effects in the final outcome of the system. The best example of this is the butterfly effect. This effect demonstrates the power of chaos theory, and the utter importance of the starting conditions. It is said that a butterfly flapping it's wings in South America may ultimately lead to a tornado appearing in North America, whereas if the butterfly did not flap it's wing, no such tornado would occur. In fact, even if the butterfly flaps it's wing just ever so slightly different, the tornado may or may not have occurred. Since the starting condition (T=0) is the butterfly flapping it's wing, that the butterfly fluttered may be said to have "caused" the tornado. Now you see why the weather is so difficult to predict, and why chaos theory is so important in understanding chaotic systems.
When I visited San Diego in December of 2003, I went to the Reuben H. Fleet Science Center. One of the exhibits demonstrated this aspect of chaos theory just perfectly. A metal pole with a (-) polarity magnet on the end hung from the ceiling. The tip with the magnet floated about 2 feet from the top of a box in the ground. Underneath the top surface of the box were 6 magnets of equal strength and all of (-) polarity, arranged in a hexagonal configuration, with each magnet equidistant from a central point in the middle of the hexagon. The magnet sat directly above this middle point at rest. Since all magnets were of like polarity, and thus would repel each other, the floating magnet would remain at rest as the magnetic fields generated from all of the magnets cancelled each other out. Painted on the top of the box was a circle divided up into 6ths, each slice of the pie with a different color.
The observer of this experiment, myself in this case, would grab the floating magnet and wind up the pendulum by pulling back to a predetermined starting position labeled X. Then, without any force exerted upon the magnet, the magnet is let go. It swings down to the circle and interacts with the magnets below them being repelled in different directions. Eventually, it stops, floating over a specific color. The point is thus: The next time I bring the magnet back to what seemingly is the exact same position as before (X), the result at the end when the pendulum stops may be a different color! This small system is displaying chaotic behavior. Even though I think I'm starting at the same exact position each and every time, if I am in actuality even a milimeter away from where I had started before, in any direction, I may get a different color as an end result. And try and try I did to get the same color even two or three times in a row and it rarely happened. It's due to the fact that the end results are predetermined by extremely precise starting conditions.
Scientifically, these effects can occasionally plunge the confirmation of theories into chaos. It is so important when conducting an experiment to precisely measure the starting conditions when chaos is in the mix. Unfortunately, we are often limited by our technological ability to measure the starting conditions so precisely, and thus, sadly, chaos takes over and the end result is ruined. Imagine an experimental hypothesis we wish to test, but where chaotic effects predominate in this testable system. Say a scientist wants to set the experiment up, and he must place a certain Object A exactly 2.56876 cm away from Object B. However, the "ruler" he uses (or other distance-measuring tool) is only able to measure up to 2.569 cm. If the experimental system is chaotic, the extra 0.00024 cm in between Object A and Object B may cause a totally different result than if he used the correct distance of 2.56876 cm. The experiment may be totally ruined!
This deviation from the "true" answer may be even more far off when considering multiplicative effects upon adding more variables into the system. For example, if we added Objects C and D into the system and they both had to be a precise distance away from Objects A and B, the experimental results may be completely off of the mark.
The infinitesimal plays a role here. I remember thinking when I was very young that no two events could ever be the exact same, because it is impossible for all of the molecules that compose a particular piece of matter to move through the exact same points in space the exact same way. It may appear grossly to look exactly the same to the naked human eye, but on a molecular level, things do not appear like that at all. They appear chaotic, yet deterministic. Luckily, the smaller you go, and more precise the measurement, the more likely you are to know the initial conditions, and thus predict the outcome.
Chaos is everywhere. The stock market, the ecomony, population growth, and the atmosphere are all common chaotic systems. In the example of weather, scientists even go so far as to try to venture inside the middle of a tornado to measure the conditions. If they can place very sensitive equipment to take as precise measurements as possible, using the calculatory abilities of a computer, they can attempt to predict the path of the tornado. Chaos theory can be used in a variety of applications in attempting to solve these types of problems, although we are in the infancy of the calculatory ability needed to accurately predict the weather and stock market. Remember, sensitivity to initial conditions!
While describing these systems as "chaotic" makes it seem as if the outcome of a system could never be determined, they can, although it is really hard to do so. Despite our efforts to predict the weather and other chaotic phenomena, these systems are actually deterministic in nature. That is, the outcome (for example, the end result at the end of a specified amount of time, say T=60) is already laid out from the moment the system either started or began to be measured (T=0) based upon the starting conditions as present between physical forces and matter. But these systems are also dynamical. It is because of this that these systems appear random or "chaotic".
Chaos theory can be summed up by reading the title of this article, as one of the most important criteria of chaotic systems is their "sensitivty to initial conditions". This concept is logical yet fascinating. It simply states that changing the starting conditions, however so slightly, can produce dramatic effects in the final outcome of the system. The best example of this is the butterfly effect. This effect demonstrates the power of chaos theory, and the utter importance of the starting conditions. It is said that a butterfly flapping it's wings in South America may ultimately lead to a tornado appearing in North America, whereas if the butterfly did not flap it's wing, no such tornado would occur. In fact, even if the butterfly flaps it's wing just ever so slightly different, the tornado may or may not have occurred. Since the starting condition (T=0) is the butterfly flapping it's wing, that the butterfly fluttered may be said to have "caused" the tornado. Now you see why the weather is so difficult to predict, and why chaos theory is so important in understanding chaotic systems.
When I visited San Diego in December of 2003, I went to the Reuben H. Fleet Science Center. One of the exhibits demonstrated this aspect of chaos theory just perfectly. A metal pole with a (-) polarity magnet on the end hung from the ceiling. The tip with the magnet floated about 2 feet from the top of a box in the ground. Underneath the top surface of the box were 6 magnets of equal strength and all of (-) polarity, arranged in a hexagonal configuration, with each magnet equidistant from a central point in the middle of the hexagon. The magnet sat directly above this middle point at rest. Since all magnets were of like polarity, and thus would repel each other, the floating magnet would remain at rest as the magnetic fields generated from all of the magnets cancelled each other out. Painted on the top of the box was a circle divided up into 6ths, each slice of the pie with a different color.
The observer of this experiment, myself in this case, would grab the floating magnet and wind up the pendulum by pulling back to a predetermined starting position labeled X. Then, without any force exerted upon the magnet, the magnet is let go. It swings down to the circle and interacts with the magnets below them being repelled in different directions. Eventually, it stops, floating over a specific color. The point is thus: The next time I bring the magnet back to what seemingly is the exact same position as before (X), the result at the end when the pendulum stops may be a different color! This small system is displaying chaotic behavior. Even though I think I'm starting at the same exact position each and every time, if I am in actuality even a milimeter away from where I had started before, in any direction, I may get a different color as an end result. And try and try I did to get the same color even two or three times in a row and it rarely happened. It's due to the fact that the end results are predetermined by extremely precise starting conditions.
Scientifically, these effects can occasionally plunge the confirmation of theories into chaos. It is so important when conducting an experiment to precisely measure the starting conditions when chaos is in the mix. Unfortunately, we are often limited by our technological ability to measure the starting conditions so precisely, and thus, sadly, chaos takes over and the end result is ruined. Imagine an experimental hypothesis we wish to test, but where chaotic effects predominate in this testable system. Say a scientist wants to set the experiment up, and he must place a certain Object A exactly 2.56876 cm away from Object B. However, the "ruler" he uses (or other distance-measuring tool) is only able to measure up to 2.569 cm. If the experimental system is chaotic, the extra 0.00024 cm in between Object A and Object B may cause a totally different result than if he used the correct distance of 2.56876 cm. The experiment may be totally ruined!
This deviation from the "true" answer may be even more far off when considering multiplicative effects upon adding more variables into the system. For example, if we added Objects C and D into the system and they both had to be a precise distance away from Objects A and B, the experimental results may be completely off of the mark.
The infinitesimal plays a role here. I remember thinking when I was very young that no two events could ever be the exact same, because it is impossible for all of the molecules that compose a particular piece of matter to move through the exact same points in space the exact same way. It may appear grossly to look exactly the same to the naked human eye, but on a molecular level, things do not appear like that at all. They appear chaotic, yet deterministic. Luckily, the smaller you go, and more precise the measurement, the more likely you are to know the initial conditions, and thus predict the outcome.
Chaos is everywhere. The stock market, the ecomony, population growth, and the atmosphere are all common chaotic systems. In the example of weather, scientists even go so far as to try to venture inside the middle of a tornado to measure the conditions. If they can place very sensitive equipment to take as precise measurements as possible, using the calculatory abilities of a computer, they can attempt to predict the path of the tornado. Chaos theory can be used in a variety of applications in attempting to solve these types of problems, although we are in the infancy of the calculatory ability needed to accurately predict the weather and stock market. Remember, sensitivity to initial conditions!
posted by J-Mart @ 7:06 PM
4 Comments:
The InformationWeek Blog
INFORMATIONWEEK WEBLOG Here you'll find observations, anecdotes, and analysis from our experienced staff of reporters and editors, with links to stories, surveys and other content that appear on InformationWeek ...
Your blog is very interesting, check out mine if you have a chance sometime!
I have a debt site/blog. It pretty much covers debt related stuff.
Can Sony really take on the iPod?
Time and again, news accounts have noted Sony's loss of the portable music to Apple Computer.
I recycle my old stuff here Auction maybe you should too
100 blogs in 100 days, day 30: The Ripple Efffect
Day 30 of 100 blogs in 100 days comes to us from Steve Harper Blog: The Ripple Effect About: "My BLOG is about creating ripples for those that cross our paths each day.
If you have time, go see my internet marketing course related site. It's nothing special but you might still find something interesting if that sort of ting interests you.
Thanks & Regards,
Don.
Ever since I watched the movie, Butterfly Effect, I wondered how true it was in the science world. I still have a difficult time comprehending the mess of chaos and determination and such, but who exactly founded this theory?
Post a Comment
<< Home