The graph demonstrates that the value of P vacillates at first, but then settles down to a steady state. This graph perfectly models our prevailing view of the universe. If there is something we should, or wish to control, such as crime, then we need to inject enough control, such as punishment, into the system so that the amount of crime and the degree of punishment reaches an acceptable, stable balance in society. That after all, is how nature does it [we smugly assume], sure the planets might wobble a bit as they circle the sun, but those wobbles cancel each other out and we get this eternally stable solar system. Samples of the numerical values to 15 decimal places using control factor 2.7 are
2.7 0.629629629629632 90
2.7 0.629629629629628 91
2.7 0.629629629629631 92
2.7 0.629629629629629 93
2.7 0.62962962962963 94
2.7 0.629629629629629 95
2.7 0.62962962962963 96
2.7 0.62962962962963 97
2.7 0.62962962962963 98
2.7 0.62962962962963 99
2.7 0.62962962962963 100
and
2.7 0.62962962962963 7990
2.7 0.62962962962963 7991
2.7 0.62962962962963 7992
2.7 0.62962962962963 7993
2.7 0.62962962962963 7994
2.7 0.62962962962963 7995
2.7 0.62962962962963 7996
2.7 0.62962962962963 7997
2.7 0.62962962962963 7998
2.7 0.62962962962963 7999
2.7 0.62962962962963 8000
The first column is the value of F (the control factor), the second column is the result, and the third column is the number of iterations (trials) of the run. So here we see that a stability that first appeared at trial 96 remains stable after 8000 trials! Well, that was pretty easy, this controlling the universe ain't so hard after all, all you gotta do is use the right key, and just to prove it let's set F to 3.1
2.7 0.629629629629632 90
2.7 0.629629629629628 91
2.7 0.629629629629631 92
2.7 0.629629629629629 93
2.7 0.62962962962963 94
2.7 0.629629629629629 95
2.7 0.62962962962963 96
2.7 0.62962962962963 97
2.7 0.62962962962963 98
2.7 0.62962962962963 99
2.7 0.62962962962963 100
and
2.7 0.62962962962963 7990
2.7 0.62962962962963 7991
2.7 0.62962962962963 7992
2.7 0.62962962962963 7993
2.7 0.62962962962963 7994
2.7 0.62962962962963 7995
2.7 0.62962962962963 7996
2.7 0.62962962962963 7997
2.7 0.62962962962963 7998
2.7 0.62962962962963 7999
2.7 0.62962962962963 8000
The first column is the value of F (the control factor), the second column is the result, and the third column is the number of iterations (trials) of the run. So here we see that a stability that first appeared at trial 96 remains stable after 8000 trials! Well, that was pretty easy, this controlling the universe ain't so hard after all, all you gotta do is use the right key, and just to prove it let's set F to 3.1
Chaos Theory for Dummies might not be available at your local book store, but the (fill in the blank) For Dummies series is growing at an enviable pace. When the series began no doubt many customers were a little reluctant to approach the cashier holding a book for dummies, but now it's cool. Early in our evolution the separation between the smarties and the dummies was minimal. It was pretty easy to demonstrate to one's peers the advantages of throwing a rock with a sling over throwing it from one's hand. Improvement in mechanical devices permitted progress from ideas to implementation homogeneously, with little need to distinguish between the one who initiated the idea and the ones who put it to use. As our ability to form abstractions began to play a role in the progress of civilization the distinction between smarties and dummies became more pronounced. The ability to imagine doing something differently gradually became more desirable than the ability to do something someone else had imagined, and so the smarties began to outrank the dummies.
As imaginations proliferated they began to form themselves into groups, which were eventually labeled disciplines. Within each discipline, fewer and fewer members were capable of imagining an extension of the boundaries, and the gap between the imaginers/smarties and the doers/dummies became far too great for direct communication. The abstract imaginary idea had to be translated through a series of simplifications until it reached the level where it could be mechanized. Once the imagination concepts were reduced to an operational system, then a second series of boundary extensions began to develop- techniques. Techniques would seem on the surface to be a bit more tangible than imaginations. Improvement in technique generally results in a corresponding improvement in performance, a simple linear progression, but just as was the case with imaginers, a few individuals were able to perform beyond the boundaries of technique development. In either case progress beyond the known boundaries cannot be mapped out ahead of time and can only be described metaphorically.
What had begun as an interactive communication process directly applicable to solving the
problem at hand evolved into two separate processes, each one capable of internal development. In each case the ability to extend the boundaries became ever more difficult, and the number of members capable of extending the boundary along either track became fewer and fewer. Soon, direct communication between the extenders of the two tracks became virtually impossible, eventually leading to competition between the tracks. As these separate processes developed the hierarchy between them began to reverse as well. The doer/dummies began to outrank the imaginer/smarties. No one would know the name of a car manufacturing engineer who used the mathematics of chaos to design a superior air flow, but everyone would know the name of the race driver that started winning more races using the design. At the end of the year the race driver would go home with a million dollar check and a prestigious trophy, while the engineer would be awarded a nice departmental commendation letter and a 5 % raise.
For a long time the multiple tracks indigenous to civilization seemed to settle into a fairly consistent cycle. Each track had a more or less proscribed developmental cycle; the entry level in each track would be the lowest rung on the hierarchical ladder, and the requirements for progress were clearly defined.
There are three primary elements from this model of persons organization important to this writing: (1) The step beyond the top step can only be described metaphorically; (2) the development of the (fill in the blank) For Dummies made it possible to jump ahead in the linear developmental process of each track, and also facilitated the ability to jump from one track to another without starting at the beginning; and (3) the organizational structure of civilization, which was thought to be periodic turns out to be aperiodic, hence unpredictable, and subject to chaos.
The multidrivium that came to be called Chaos Theory had its roots in mathematics, and one could hardly write about Chaos Theory without reference to its mathematical development. But before we step into the hallowed halls of mathematics and risk getting tossed out on our ears, we need, "a ticket to ride", and we can earn that by matching wits with an expert and devising a superior explanation.
Background. Previous to moving to the beach, I lived in the mountains west of Boulder. Several homes shared a dirt driveway leading to a public road. The driveway ran basically east to west. The prevailing winds, which could be quite fierce, came from the west. We experienced an invasion by a noxious weed, the thistle. There were spots here and there in the landscape where it became the predominant plant, crowding out the more desirable vegetation. Occasionally, residents hiking along the road would pull up the offending plants growing along the side of the driveway, both when they were immature as well as after they had developed seeds. They would toss these uprooted plants in the middle of the driveway where traffic and lack of good soil would prevent their germination. Despite these efforts the thistles made steady progress down the driveway, they would predominately sprout within a foot or two of the edge of the driveway, but each spring new plants would sprout well beyond where the last of the thistles had grown the year before.
I had a neighbor who would spend his summers in his mountain home near mine. During the rest of the year he held a distinguished professor of mathematics position at a prestigious ivy league university. He was widely considered by his peers to be one of the top mathematicians in the world, and while his work was published frequently it was never published in large volume, simply because the body of readers who could understand his work was very small. He and his wife would frequently ride their horses along the driveway and into the forest. Since I hiked the same trails we would occasionally cross paths and chat about the things neighbors talk about. One day we were discussing the march of the thistles and he explained to me why their progress was greatest along the edge of the driveway. It was a matter of competition, he reasoned. The density of plants immediately adjacent to the edge of the driveway was less than the density of plants in a comparable area away from the driveway's edge, hence the thistles were able to expand their progress only where the density of competing plants was less, but unable to propagate as easily away from the driveway where the local vegetation held sway. Well, he was wrong. Your assignment is to come up with a better explanation for the insidious growth pattern of the evil thistles.
[ Cut and paste your answer here.]
The purpose of this simple-minded example is not just to reinforce the well known fact that expertise in one field does not make one an expert in other fields, but to pave the way for all of us to recognize that the deference we have traditionally given to the status of experts in general is gradually receding. (Fill in the blank) For Dummies, and chaos theory have not only given us the tools to question the experts, but also the responsibility. First, a brief look at the mathematical genesis of Chaos Theory.
While elements of Chaos Theory can be traced back a long way and through many persons, it is generally accepted that most of the blame can fall on Edward Lorenz. Lorenz was playing on his new Royal McBee computer in the early 1960's with the goal of developing a program to predict the weather. Lorenz' expectations of success were high given his exceptional mathematical background and his new toy. In many ways, the computer promised to be the holy grail of theoretical science. The level of computations required to solve many of the scientific problems would often exceed the ability of a scientist to complete the calculations in a single lifetime. [The computational speed of Edward's McBee was probably less than the average Casio wrist watch today, and even though super computers now have speeds way too fast to count, the holy grail expectations have drifted away.] Lorenz' scientific background was tilted heavily toward the Newtonian view of existence. As we shall see, most of the ways we organize ourselves, and even many of our values have been formed with Mr. Isaac Newton looking over our shoulders. This deterministic view of the universe holds that the universe runs on a conglomeration of immutable natural laws. The scientist's role was to discover the mathematical basis of these laws. In addition, the interaction between these natural laws (the acceleration of gravity, the speed of light, stuff like that) was also based on a series of laws that could be understood mathematically. Eventually, it was expected that we would be able to successfully predict and even control the clockwork fate of the universe based on these interactive equations. This school of thought has had a remarkable record of success, imagine the idea of performing a bunch of calculations that when applied to raw materials enabled us to put a man on the moon. There were of course glitches here and there in this process- a rocket might not land precisely where the mathematics predicted it would, but close enough for government work, as they say. In the long run, these little anomalies were thought to cancel each other out and not affect the larger picture.
Lorenz had developed a very simple weather model based on the interaction of various elements of weather, temperature, humidity, wind direction, etc. The model was expected to show how these variables interacted over time, thus predicting the weather pattern for an extended period. During one run of his program, while copying the start data from his previous run, he rounded off a set of numbers from six decimal places to three, an absolutely inconsequential change according to then current scientific investigative procedures because experiments cannot be performed unless very small influences are neglected. Lorenz expected the print out of the run to duplicate the previous run, assuring him that he had developed the right equations to describe the interaction of the elements. He was astonished to see the results. The weather patterns followed the previous pattern only briefly, soon however, the chart deviated slightly from the previous run, then the deviation increased rapidly. Suddenly the weather patterns between the two runs showed no resemblance to each other at all.
Irrespective of the seemingly inconsequential changes in the system, if it did not repeat exactly, in every respect, it rapidly became unpredictable. Eventually this phenomenon came to be called the "butterfly effect" from one scientist's wry observation that the flapping of a butterfly's wings in Kansas could potentially have a measurable effect on the weather in the South Pacific. [Perhaps some day some compulsive individual will track down all the examples of the butterfly effect on record to determine how many different places the butterfly is suggested as being and where its wing flapping ends up impacting the weather.] The more useful term for this consequence is extreme sensitivity to initial conditions.
Much to his disappointment Lorenz realized that his goal of making accurate long range weather predictions was impossible. I will call this realization the Lorenz Moment, and we will visit it later. At the time there was very limited communication and interaction between the various scientific disciplines, and any significant theoretical breakthroughs were expected from fields such as physics or astronomy, certainly not from the weatherman, so the implications of Lorenz' work didn't exactly stop the presses immediately. Lorenz was a weatherman only by the good graces of the fickle finger of fate. His true interest was in mathematics. Had Lorenz stopped work when he realized that accurate long range weather forecasts were impossible because the exact initial conditions could not be duplicated, nothing much would have changed. But Lorenz sensed that something much more extraordinary was going on. There seemed to be some sort of hidden order within the disorder of unpredictability.
He began to test his ideas by developing the most simple nonlinear equation he could. He settled on an equation with three variables that were thought to be related but not directly. He then sought to represent the relationship between the variables at a given point of time as a single dot, recording the results on a three dimensional graph. A line drawn from one dot to the next traced a path of the points along the three dimensions. What he found was that the path of the dots never retraced its own path, although it would almost do so. But then neither did it ever completely fly off the graph. Whether the picture that emerged actually resembled butterfly wings is open to question, but it was clear that some kind of self attracting, and self limiting, pattern was occurring in the system. Lack of interdisciplinary communication was not the only reason Lorenz' work did not garner much attention, perhaps even more importantly, the implications of his findings were simply a pill too big to swallow at once.
Skipping over a list of distinguished contributors by using the awesome powers of editorial discretion, the research efforts sought to find the simplest way to develop predictive models that accurately depicted real life. Obviously the more complex and multifaceted a predictive model, the greater the chances are that something will go wrong. In terms of scientific experimentation that means reducing the number of variables. It is much simpler to do a modest experiment that is fail safe, and then gradually add one variable at a time, than to start with large number of variables, and then try to figure out what went wrong (or right). The formula had to be based on a single element influencing the status of another where the relationship was not linear, that is that the value of one element did not change proportionally with the value of another, such as would be the case if one began stacking a pile of 5 pound weights on a scale. Assuming accuracy of the scale each 5 pound weight would move the dial showing the weight an equal distance. The model also needed to be self limiting, the process of stacking weights on a scale would have to end at some point, the scale would break or the weights would topple over, for example. Similarly, if all the weights were removed from the scale, the dial would register zero. Researchers developed an iterative equation initially conceived for modeling populations, where the population after one calculation was used as the start value for the next calculation. An equation frequently used to demonstrate this process goes
P(n) = FP (1-P)
Where P(n) is the result of the previous iteration using F as the hypothetical element driving the value of P, as the hypothetical value of the thing to be estimated, and (1-P) which keeps the results between 1 and zero. That was thought to be simple enough to provide accurate estimates of future (whatever was being predicted). The value of F would be manipulated until the value of P(n) stabilized, hopefully at or near the value predicted by the researcher, at which point the researcher would publish her/his results and wait for the research grants to come pouring in. [Shareware is readily available on the net to demonstrate this as well as vastly more complicated interactions. Most of the programs are capable of producing eerily self similar graphics such as Mandelbrot sets.] But for our purposes, the complexity of chaos can be demonstrated quite simply with a calculator. If you have a spreadsheet on your computer, you can set up the program yourself and follow along, and then play by yourself when class is over.
Set the P value in column A, row 1, to .04, (or any arbitrary value between 0-1)
Set column B, row 1 at (1 minus column A, row 1).
Set column C, row 1 at ( Column A, row 1 times Column B, row 1)
Set column D at 2.7 ( This is our arbitrary control factor)
Set column E at (Column C, row 1 times column D, row 1).
Set column A, row 2 to the value at Column E, row 1.
And off we go!
A VERY SCIENTIFIC EXPERIMENT
When we transfer our iterations to a graph we get
Things look a little different here. The value of P wandered around a bit just as it did in the 2.7 run. But then the graph began showing a bifurcation (the output alternates between one set of values going progressively up and another set of values that seems to be going down). And the numbers are
3.1 0.558014125202698 160 3.1 0.764566519958594 161 3.1 0.558014125202697 162 3.1 0.764566519958594 163 3.1 0.558014125202697 164 3.1 0.764566519958594 165 3.1 0.558014125202696 166 3.1 0.764566519958594 167 3.1 0.558014125202696 168 3.1 0.764566519958594 169 3.1 0.558014125202696 170
and
3.1 0.558014125202696 7990 3.1 0.764566519958594 7991 3.1 0.558014125202696 7992 3.1 0.764566519958594 7993 3.1 0.558014125202696 7994 3.1 0.764566519958594 7995 3.1 0.558014125202696 7996 3.1 0.764566519958594 7997 3.1 0.558014125202696 7998 3.1 0.764566519958594 7999 3.1 0.558014125202696 8000
In this trial the value of P wandered around as usual until iteration 165 when it split into the bifurcated values (also called period doubling). Nonetheless this two value result is itself stable as shown in trials 7999 and 8000. That's still predictable— when the value ...8594 shows up we know that the next value will be ...2696. But when we set the F value to only .6 higher at 3.7...
3.1 0.558014125202698 160 3.1 0.764566519958594 161 3.1 0.558014125202697 162 3.1 0.764566519958594 163 3.1 0.558014125202697 164 3.1 0.764566519958594 165 3.1 0.558014125202696 166 3.1 0.764566519958594 167 3.1 0.558014125202696 168 3.1 0.764566519958594 169 3.1 0.558014125202696 170
and
3.1 0.558014125202696 7990 3.1 0.764566519958594 7991 3.1 0.558014125202696 7992 3.1 0.764566519958594 7993 3.1 0.558014125202696 7994 3.1 0.764566519958594 7995 3.1 0.558014125202696 7996 3.1 0.764566519958594 7997 3.1 0.558014125202696 7998 3.1 0.764566519958594 7999 3.1 0.558014125202696 8000
In this trial the value of P wandered around as usual until iteration 165 when it split into the bifurcated values (also called period doubling). Nonetheless this two value result is itself stable as shown in trials 7999 and 8000. That's still predictable— when the value ...8594 shows up we know that the next value will be ...2696. But when we set the F value to only .6 higher at 3.7...
Continued
Everyday Chaos Theory
