Lorenzs Work in the Chaos Field and Basic Chaos

Edward Lorenz was a mathematical meteorologist during the 1960s. In 1961, an experiment with a primitive weather predicting program lead to the discovery of the theory of chaos. Lorenz defined chaos as “a system that has two states that look the same on separate occasions, but can develop into states that are noticeably different. He started exploring further into the chaos field and performing experiments that lead to his discovery of the Lorenz equations in 1963.

In 1961, Lorenz developed a weather model consisting of twelve non-linear equations. This model included barometric pressure, wind velocity, temperature, etc.. After running it on his computer, it seemed to give good results. He ran this model many times with different starting variables in each equation to see how it behaved and if it did follow the model of real weather. One time, after he completed a particularly long weather sequence, he decided to let it run longer and to start the program over again at the previous sequences mid-point. He entered the information and supposedly went for coffee. When he returned, he was confused to find results of the beginning of the new sequence not matching up with the results of the middle of the last run. The numbers were not very close and growing farther apart as the sequence progressed. Lorenz then thought there was a bug in the system. After much double checking, he found the problem. When he had entered the data into the second run, he had shortened one decimal. He had cut 0.506127 at 0.206 thinking it would not make a difference. Having

understood the importance of one thousandth of a part, Lorenz had understood the basis for a chaotic, non-linear system.

Another very important experiment of Lorenzs is known as the dueling calculators. Lorenz used two calculators to perform the same iteration. Calculator one had ten significant digits, while calculator two had twelve.

The resulted of this experiment are below.

calculation numbercalc1calc2

10.03970000000.039700000000

20.15407173000.154071730000

30.54507262600.545072626044

41.28897800101.288978001190

50.17151914210.171519142100

100.72291430120.722914301711

200.59652924470.596528770927

300.37420923210.3746447695060

401.21976311501.230600865510

500.00366162950.225758993339

After 3- 5 iterations, one minor difference is noted in the eleventh place. However, after 50 iterations, the answers are completely different.

The Lorenz equations were discovered by Ed Lorenz in 1963 as a very simplified model of convection rolls in the upper atmosphere. Later these same equations appeared in studies of lasers, batteries, and in a simple chaotic waterwheel.

Lorenz found that the trajectories of this system, for certain settings, never settle down to a fixed point, never approach a stable limit cycle, yet never diverge to infinity. What Lorenz discovered was unheard of in the mathematical community, and ignored for many years.

One of the most simple physical models of one of the Lorenz equations is a rotating waterwheel. The flow of water into the top cup, which pours into the next, and so on to keep the wheel rotating. If the water flow is too slow, the water leaks out too fast and friction prevents the wheel from rotating. If the water flow is increased a little, the wheel will rotate in one direction forever. If the flow is too fast, then the wheel will not settle into a stable cycle. The wheel will then spin in one direction, then slow down, stop, and start spinning in the other direction. This process will continue infinitely, but without regularity or pattern.

Chaos is defined as Stochastic behavior in deterministic systems, or in laymans terms, Ruleless behavior is governed by rules. Chaos is also the study of nonlinear dynamics. Dynamics sensitive to their initial conditions; if the conditioned change, even a very little bit, the entire equation will be changed a great deal.

Chaotic Systems appear random, but have three defining characteristics.

The first trait is determinability. Something is determining their behavior. Second, they are exceptionally sensitive to initial conditions; one minor change in the beginning leads to a completely different outcome. Third, there exists an orderly sense within all chaotic systems. In fact, truly random systems are not chaotic because they do not have even a slight pattern.

With the rapid new discoveries in the field of Chaos, many old ideas had to be cast out. These new chaotic ideas teach us that Newton and almost all pre-chaos scientists were incorrect in their conclusions of the Universe. Many believe there was a predictable cause and effect system incorporating everything. They also believe everything happened according to physical laws and algebra. The chaos theory introduces the idea of probability. Old science put their trust in certainty; everything was possible to predict if you knew all of the initial conditions. Chaos teaches this is not true. It is impossible to know all the initial conditions because these conditions are constantly changing and there is no way to predict everything. Chaotic systems exist everywhere.

After careful study by respective authorities, the stock market has been declared a chaotic system, since it is a non-linear dynamic system. Much analysis of the market and prices has given way to the conclusion that stock prices are very random, but contain a trend. The strength of this trend does vary, but still exists in all markets and time frames.

The concept of fractals is very closely related to the chaos theory and chaotic systems. Fractals are objects similar to themselves and to the corresponding pieces of themselves. This fractal structure exists in the market. If one was to study the monthly, weekly, daily, and intra day stock prices, one would find the same thread of similarity running through them all.

Sensitive dependence on initial conditions shows up in the stock market, and rightly so. The market is a chaotic system. This trait makes all chaotic systems difficult, or close too impossible to predict. This is because no one can include all of the initial conditions with any sort of accuracy. There are too many nuances in the conditions. Even if a person could predict what was going to happen in the stock market with accuracy tomorrow, in a few days, their predictions would be completely wrong.

One odd characteristic of the stock market, is that short term data (for instance, a five minute report) is just random stock trading and longer term data, (like weekly or daily) is not. This paradox exists but cannot be fully understood yet.

Lorenzs experiments and equations have discovered the chaos theory, explored it, but have not even begun to understand it. Still unable to be proved as fact, chaos theory still needs to be explored further by specialists in the field.