Lorenz’s Work in the Chaos Field and Basic Chaos

Jessica Shaw

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 sequence’s 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 Lorenz’s 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 number calc1 calc2
1 0.0397000000 0.039700000000
2 0.1540717300 0.154071730000
3 0.5450726260 0.545072626044
4 1.2889780010 1.288978001190
5 0.1715191421 0.171519142100
10 0.7229143012 0.722914301711
20 0.5965292447 0.596528770927
30 0.3742092321 0.3746447695060
40 1.2197631150 1.230600865510
50 0.0036616295 0.225758993339

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 layman’s 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.