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
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...
the path of P becomes chaotic. Checking back with our numbers we see that even after 8000 iterations no steady state has emerged. We are looking at chaos, there is simply no way to predict the next trial result from looking at the present one.
3.7 0.591341613085392 7990
3.7 0.894129825960147 7991
3.7 0.350248217067911 7992
3.7 0.842025292982044 7993
3.7 0.492169186154025 7994
3.7 0.924773109911686 7995
3.7 0.257400878855033 7996
3.7 0.707238965752851 7997
3.7 0.766092440972648 7998
3.7 0.663020807571706 7999
3.7 0.826669600305074 8000
Well, this was not the thing the reductionist folks wanted to hear. Although there had been prior warnings, the uncertainty principle in quantum mechanics for example, pretty much the heart and soul of all science was bent on finding final explanations. If we kept looking, just kept our noses to the scientific method grindstone, we would eventually get to the part that said 1+1= 2. The prospect of there not being a real, unchanging bottom of things; that a flake of dandruff falling from an angel's shoulder could upset the universal apple cart is just contrary to the nature of our species. At this point it was clear to those who looked closely that one of the basic assumptions
in scientific research was flawed. Tiny variations in a system were not necessarily absorbed as time went on. Not every little change disrupted the system, but it had the potential to. The implications of these findings caused barely a ripple in the course of scientific research, and they were largely ignored by the rest of us, but it did attract more attention to the boundary between order and chaos. If nothing in a steady state flow of data indicated that chaos was approaching, and nothing in chaos explained how it got there, perhaps the apparently benign dividing line between order and chaos was more complicated than previously thought.
A few researchers noticed that something unusual did seem to be operative in the boundary between order and chaos. They observed that in the complex areas of the graphs they were producing, a series of events took place that came to be called period doubling. A steady data stream would break into two streams, then suddenly jump to four streams, then 8, then 16... In graphs depicting chaos, there seemed to be brief little periods of seeming order that would appear, quickly disappear, only to reveal other orderly sections as time went on. The simple formulas, left to act on their own over time (by iterations), began to produce results that made no sense from understanding how the formula elements related to each other. For the most part researchers treated these findings much like a mathematical curiosity, interesting but not really helpful to their particular line of research. Although he was ostensibly involved in unrelated research, something about the boundary between order and disorder intrigued Mitchell Feigenbaum. Feginbaum became interested, some would say obsessed, with understanding exactly what was going on during the period when, for example steady states such as illustrated in our 2.7 graph were changing to the state illustrated in the 3.1 graph. Was some similar "thing" going on in the boundary between the states of things in the 3.1 graph and the 3.7 graph?
Mitchell's big breakthrough came with the realization that while the relationship seemed to be invisible numerically, it could be seen geometrically. In a data stream bound for chaos, the first bifurcation of two related but different data streams would be followed by a period of chaotic data, only to split again into 4 streams of data. As these data streams continued to double, the period between bifurcations became shorter and shorter. Perceptually, if one looks down a railroad track, the tracks seem to grow closer and closer together as the distance increases, and Mitchell realized that the same thing was happening to the spaces (iteration sequences) between period doublings. Geometric convergence meant that something orderly was being preserved while everything around it was becoming chaotic.
Feigenbaum initially calculated that bifurcation doubling occurred at a ratio that approached the irrational number 4.669 [when he got access to a better computer he was able to calculate the number out to 4.66921160910299067185320382047...]. The amazing thing Feigenbaum found was that the constant value applied to other formulas that exhibited period doubling characteristics. Even when he moved from simple one dimensional equations describing animal populations to complex differential equations describing the transition to turbulence in a stream of liquid, he found the same ratio applied to the time between period doublings. Indeed Feigenbaum had developed a universal constant that now bears his name. joining the esteemed ranks of constants right along with pi and the golden ratio. The path to chaos was not an instantaneous random leap from one state to another, but a well ordered, progressive transition from the expected path, following ever more frequent precision doublings until the data trail disappeared into the tangled mess of chaos.
The Feigenbaum constant plays an essential role in "real life", not just in the classroom.
The constant has influenced study of the human heart, electronic oscillators, chemical reactions, lasers, fluid-flow turbulence--- the list is constantly expanding. [ pun initially unintentional, but I decided to take credit for it anyway.] If we truly understood its impact, we might rate it right up there with gravity. While the mathematical exploration of chaos continues apace, we can drop out of the chase at this point because we have learned a vital part of what we need to know- and that is that predictions can never be a fact certain. While Murphy's Law that if anything can go wrong it will, fortunately doesn't typically apply, it is true that anything can go wrong, and all to often the culprit turns out to be some apparently inconsequential thing we scarcely noticed. While we can't clearly demonstrate Feigenbaum's constant in ordinary events, we can imagine it.
Picture a human pyramid balanced on a high wire, perhaps the famous Mendazzle family in Joe's Circus. As the climax of their act four members of the family walk out on the high wire blindfolded [of course the blindfolds are not necessary, but it's my example]. Then three members, also blindfolded, walk out and climb up on the shoulders of their siblings. Then two on top of their shoulders. Finally little Lucy Mendazzle climbs over her family and stands on top of the pyramid. Now they all lift one arm and paste on the mandatory plastic smile. The audience claps enthusiastically as the Mendazzle family stands there motionless. But let's shift our perspective, rather than view them as 10 people, think of them as one musculature structure in constant, almost imperceptible motion. Thousands of muscles are flexing and relaxing in a coordinated effort to maintain a steady state system that took years of practice to perfect. But suddenly little Lucy feels like she has to sneeze. She manages to stifle the sneeze, but the effort causes her concentration to waver slightly. She shifts her weight a tiny bit more to her right leg. Jeff and John feel the slight change, and shift their stance unconsciously. The pyramid still appears steady as a rock, but the slight changes are unexpected, they are out of the usual rhythm. The out of sequence perturbation originating in Lucy's leg muscles creates a cascading plethora of out of balance motions that multiply in intensity as it travels down muscle connections to the wire. The crowd senses that something is going wrong, something just looks different. Before anyone can show outward concern the pyramid comes crashing down and they all die. We have witnessed period doubling leading to chaos, and somehow, we perceived it before it happened.
The extreme sensitivity to initial conditions concept has not made its way into our everyday view of reality- but it is real. The only way we have learned to look at it is in retrospect. The NTSB pores over the every tiny detail of a plane crash for months, even years before they can understand the single little thing that started the disaster sequence. But if nothing is perfectly predictable, if everything orderly is one tiny misstep away from disaster, how can we possibly continue? If chaos theory was only about the way all structure is subject to disintegration, it might as well be the Second Law of Thermodynamics.
But never fear, here come de chaos fairy. Look at the numbers produced by an F of 3.7. They just don't make sense. Maybe a big computer that could do a kazillion iterations out to a million decimal places might find numerical sequences there, but that could be done only by the elite of the smartie/imaginers, and even then we wouldn't be able to put the numbers to practical use- assuming of course that we understand what they were talking about. Rather than flail away at our keyboard, let's try another approach. Look back at the graph for F 3.7. See anything odd; like patterns; like something in there is trying to repeat itself but just can't get it right? [The almost patterns can be demonstrated numerically, but not on my spreadsheet. Besides even the big guys eventually have to get off the numbers train.] What we are seeing is self similarity, the basis of fractals, of disorder seeking to become orderly, but on a different scale.
And that brings us to my favorite chaos theory term- the strange attractor. Mathematicians and scientists in general can be pretty persnickety about the definition of a strange attractor. Part of the reason behind these linguistic turf wars lies in the fact that we all have a reasonable idea about the meaning of both terms. [For this, the science guys have no one to blame but themselves- if they had stuck to Latin to name stuff rather than try to take over "our" words for themselves, there would be no controversy.] But for us, the goal is not to define strange attractors, but to recognize them; to use them as tools to help us make decisions.
Scientific researchers recognize several kinds of attractors, but for our purposes we just need a brief look, enough to give us a feel for the concept. Shamelessly lifting an example used by others, let's do another demonstration. Imagine a stainless steel bowl of acceptable size and shape resting on a table. Take a steel ball bearing and hold it in your forefinger with your thumbnail resting against the side of the ball. Then in a motion familiar to anyone who has played marbles, flick the ball bearing forward just under the rim of the bowl. The speed is not critical as long as you don't flip it completely out of the bowl. You have a handy list of equations to enable you to quantify everything that happens in the bowl, which you can then depict graphically. [The system within the bowl is called the phase space, and we will keep it in the back of our minds in case we need to refer to it later. Basically the phase space contains the set of all possible states of the system.] A number of factors (momentum, etc.) affect the trajectory of the ball around the bowl, but eventually it will come to rest at the bottom of the bowl. This is called a point attractor, since the final resting place pulls or "attracts" the trajectories of the ball to that spot.
Graphical records of the course of more complex systems might show different attractors, but the view of reductionist science held that all systems, given the correct equations and the computational power of computers would eventually be found to have an underlying repeatability, hence predictability governing the system. The Lorenz Attractor upset that apple cart but it didn't [ and I can't believe I am actually typing this, waste the apples. Continuing with my little digression to disguise my difficulty in composing the next sentence, what if Newton had been 100% correct? Suppose we lived in a world where everything was perfectly understood and under control. How safe. How secure. How goddam boring ! ] Chaotic behavior of a system is not random behavior, it just looks that way from one perspective- that of knowing exactly what will happen next. Nor is a chaotic system one without boundaries. It may be pushing the metaphor a little too far, but it's like real systems (probably all of them) are held together rather loosely by strange attractors. The area of phase space under the influence of a given attractor is known as the basin of attraction. While mathematics and computers were the tools that brought strange attractors to our attention, we have been living with, and using them for a long time. Observe one for yourself. Assume you live in the lower 48 north of the Mason Dixon line. Step outside your front door with a thermometer in your hand. You check the temperature and it is 94 degrees Fahrenheit. Go back inside and come out and recheck the temperature in exactly 30 seconds. Using your innate strange attractor detection skills, you can confidently say the following kinds of things: It's summertime; the temperature 30 seconds from now will be 94 degrees fahrenheit; I did not put on my snow boots before stepping outside; the dead of winter will be here within 6 months and the average temperature during one week of that period will be at least 30 degrees cooler than it is now; it is not possible for me to know what the temperature will be at this time tomorrow, but I know it will be under 140 degrees (if it is hotter than that when I step outside tomorrow I should keep moving because it probably means my house is on fire).
When we formed our prevailing view of the universe one of the initial conditions we assumed was that immutable laws governed the clockwork machinations and we would gradually take control of our destiny by discovering those laws. The sensitivity of reality is responding to that initial assumption, and even now the impact of our error is gaining strength in the background.
3.7 0.591341613085392 7990
3.7 0.894129825960147 7991
3.7 0.350248217067911 7992
3.7 0.842025292982044 7993
3.7 0.492169186154025 7994
3.7 0.924773109911686 7995
3.7 0.257400878855033 7996
3.7 0.707238965752851 7997
3.7 0.766092440972648 7998
3.7 0.663020807571706 7999
3.7 0.826669600305074 8000
Well, this was not the thing the reductionist folks wanted to hear. Although there had been prior warnings, the uncertainty principle in quantum mechanics for example, pretty much the heart and soul of all science was bent on finding final explanations. If we kept looking, just kept our noses to the scientific method grindstone, we would eventually get to the part that said 1+1= 2. The prospect of there not being a real, unchanging bottom of things; that a flake of dandruff falling from an angel's shoulder could upset the universal apple cart is just contrary to the nature of our species. At this point it was clear to those who looked closely that one of the basic assumptions
in scientific research was flawed. Tiny variations in a system were not necessarily absorbed as time went on. Not every little change disrupted the system, but it had the potential to. The implications of these findings caused barely a ripple in the course of scientific research, and they were largely ignored by the rest of us, but it did attract more attention to the boundary between order and chaos. If nothing in a steady state flow of data indicated that chaos was approaching, and nothing in chaos explained how it got there, perhaps the apparently benign dividing line between order and chaos was more complicated than previously thought.
A few researchers noticed that something unusual did seem to be operative in the boundary between order and chaos. They observed that in the complex areas of the graphs they were producing, a series of events took place that came to be called period doubling. A steady data stream would break into two streams, then suddenly jump to four streams, then 8, then 16... In graphs depicting chaos, there seemed to be brief little periods of seeming order that would appear, quickly disappear, only to reveal other orderly sections as time went on. The simple formulas, left to act on their own over time (by iterations), began to produce results that made no sense from understanding how the formula elements related to each other. For the most part researchers treated these findings much like a mathematical curiosity, interesting but not really helpful to their particular line of research. Although he was ostensibly involved in unrelated research, something about the boundary between order and disorder intrigued Mitchell Feigenbaum. Feginbaum became interested, some would say obsessed, with understanding exactly what was going on during the period when, for example steady states such as illustrated in our 2.7 graph were changing to the state illustrated in the 3.1 graph. Was some similar "thing" going on in the boundary between the states of things in the 3.1 graph and the 3.7 graph?
Mitchell's big breakthrough came with the realization that while the relationship seemed to be invisible numerically, it could be seen geometrically. In a data stream bound for chaos, the first bifurcation of two related but different data streams would be followed by a period of chaotic data, only to split again into 4 streams of data. As these data streams continued to double, the period between bifurcations became shorter and shorter. Perceptually, if one looks down a railroad track, the tracks seem to grow closer and closer together as the distance increases, and Mitchell realized that the same thing was happening to the spaces (iteration sequences) between period doublings. Geometric convergence meant that something orderly was being preserved while everything around it was becoming chaotic.
Feigenbaum initially calculated that bifurcation doubling occurred at a ratio that approached the irrational number 4.669 [when he got access to a better computer he was able to calculate the number out to 4.66921160910299067185320382047...]. The amazing thing Feigenbaum found was that the constant value applied to other formulas that exhibited period doubling characteristics. Even when he moved from simple one dimensional equations describing animal populations to complex differential equations describing the transition to turbulence in a stream of liquid, he found the same ratio applied to the time between period doublings. Indeed Feigenbaum had developed a universal constant that now bears his name. joining the esteemed ranks of constants right along with pi and the golden ratio. The path to chaos was not an instantaneous random leap from one state to another, but a well ordered, progressive transition from the expected path, following ever more frequent precision doublings until the data trail disappeared into the tangled mess of chaos.
The Feigenbaum constant plays an essential role in "real life", not just in the classroom.
The constant has influenced study of the human heart, electronic oscillators, chemical reactions, lasers, fluid-flow turbulence--- the list is constantly expanding. [ pun initially unintentional, but I decided to take credit for it anyway.] If we truly understood its impact, we might rate it right up there with gravity. While the mathematical exploration of chaos continues apace, we can drop out of the chase at this point because we have learned a vital part of what we need to know- and that is that predictions can never be a fact certain. While Murphy's Law that if anything can go wrong it will, fortunately doesn't typically apply, it is true that anything can go wrong, and all to often the culprit turns out to be some apparently inconsequential thing we scarcely noticed. While we can't clearly demonstrate Feigenbaum's constant in ordinary events, we can imagine it.
Picture a human pyramid balanced on a high wire, perhaps the famous Mendazzle family in Joe's Circus. As the climax of their act four members of the family walk out on the high wire blindfolded [of course the blindfolds are not necessary, but it's my example]. Then three members, also blindfolded, walk out and climb up on the shoulders of their siblings. Then two on top of their shoulders. Finally little Lucy Mendazzle climbs over her family and stands on top of the pyramid. Now they all lift one arm and paste on the mandatory plastic smile. The audience claps enthusiastically as the Mendazzle family stands there motionless. But let's shift our perspective, rather than view them as 10 people, think of them as one musculature structure in constant, almost imperceptible motion. Thousands of muscles are flexing and relaxing in a coordinated effort to maintain a steady state system that took years of practice to perfect. But suddenly little Lucy feels like she has to sneeze. She manages to stifle the sneeze, but the effort causes her concentration to waver slightly. She shifts her weight a tiny bit more to her right leg. Jeff and John feel the slight change, and shift their stance unconsciously. The pyramid still appears steady as a rock, but the slight changes are unexpected, they are out of the usual rhythm. The out of sequence perturbation originating in Lucy's leg muscles creates a cascading plethora of out of balance motions that multiply in intensity as it travels down muscle connections to the wire. The crowd senses that something is going wrong, something just looks different. Before anyone can show outward concern the pyramid comes crashing down and they all die. We have witnessed period doubling leading to chaos, and somehow, we perceived it before it happened.
The extreme sensitivity to initial conditions concept has not made its way into our everyday view of reality- but it is real. The only way we have learned to look at it is in retrospect. The NTSB pores over the every tiny detail of a plane crash for months, even years before they can understand the single little thing that started the disaster sequence. But if nothing is perfectly predictable, if everything orderly is one tiny misstep away from disaster, how can we possibly continue? If chaos theory was only about the way all structure is subject to disintegration, it might as well be the Second Law of Thermodynamics.
But never fear, here come de chaos fairy. Look at the numbers produced by an F of 3.7. They just don't make sense. Maybe a big computer that could do a kazillion iterations out to a million decimal places might find numerical sequences there, but that could be done only by the elite of the smartie/imaginers, and even then we wouldn't be able to put the numbers to practical use- assuming of course that we understand what they were talking about. Rather than flail away at our keyboard, let's try another approach. Look back at the graph for F 3.7. See anything odd; like patterns; like something in there is trying to repeat itself but just can't get it right? [The almost patterns can be demonstrated numerically, but not on my spreadsheet. Besides even the big guys eventually have to get off the numbers train.] What we are seeing is self similarity, the basis of fractals, of disorder seeking to become orderly, but on a different scale.
And that brings us to my favorite chaos theory term- the strange attractor. Mathematicians and scientists in general can be pretty persnickety about the definition of a strange attractor. Part of the reason behind these linguistic turf wars lies in the fact that we all have a reasonable idea about the meaning of both terms. [For this, the science guys have no one to blame but themselves- if they had stuck to Latin to name stuff rather than try to take over "our" words for themselves, there would be no controversy.] But for us, the goal is not to define strange attractors, but to recognize them; to use them as tools to help us make decisions.
Scientific researchers recognize several kinds of attractors, but for our purposes we just need a brief look, enough to give us a feel for the concept. Shamelessly lifting an example used by others, let's do another demonstration. Imagine a stainless steel bowl of acceptable size and shape resting on a table. Take a steel ball bearing and hold it in your forefinger with your thumbnail resting against the side of the ball. Then in a motion familiar to anyone who has played marbles, flick the ball bearing forward just under the rim of the bowl. The speed is not critical as long as you don't flip it completely out of the bowl. You have a handy list of equations to enable you to quantify everything that happens in the bowl, which you can then depict graphically. [The system within the bowl is called the phase space, and we will keep it in the back of our minds in case we need to refer to it later. Basically the phase space contains the set of all possible states of the system.] A number of factors (momentum, etc.) affect the trajectory of the ball around the bowl, but eventually it will come to rest at the bottom of the bowl. This is called a point attractor, since the final resting place pulls or "attracts" the trajectories of the ball to that spot.
Graphical records of the course of more complex systems might show different attractors, but the view of reductionist science held that all systems, given the correct equations and the computational power of computers would eventually be found to have an underlying repeatability, hence predictability governing the system. The Lorenz Attractor upset that apple cart but it didn't [ and I can't believe I am actually typing this, waste the apples. Continuing with my little digression to disguise my difficulty in composing the next sentence, what if Newton had been 100% correct? Suppose we lived in a world where everything was perfectly understood and under control. How safe. How secure. How goddam boring ! ] Chaotic behavior of a system is not random behavior, it just looks that way from one perspective- that of knowing exactly what will happen next. Nor is a chaotic system one without boundaries. It may be pushing the metaphor a little too far, but it's like real systems (probably all of them) are held together rather loosely by strange attractors. The area of phase space under the influence of a given attractor is known as the basin of attraction. While mathematics and computers were the tools that brought strange attractors to our attention, we have been living with, and using them for a long time. Observe one for yourself. Assume you live in the lower 48 north of the Mason Dixon line. Step outside your front door with a thermometer in your hand. You check the temperature and it is 94 degrees Fahrenheit. Go back inside and come out and recheck the temperature in exactly 30 seconds. Using your innate strange attractor detection skills, you can confidently say the following kinds of things: It's summertime; the temperature 30 seconds from now will be 94 degrees fahrenheit; I did not put on my snow boots before stepping outside; the dead of winter will be here within 6 months and the average temperature during one week of that period will be at least 30 degrees cooler than it is now; it is not possible for me to know what the temperature will be at this time tomorrow, but I know it will be under 140 degrees (if it is hotter than that when I step outside tomorrow I should keep moving because it probably means my house is on fire).
When we formed our prevailing view of the universe one of the initial conditions we assumed was that immutable laws governed the clockwork machinations and we would gradually take control of our destiny by discovering those laws. The sensitivity of reality is responding to that initial assumption, and even now the impact of our error is gaining strength in the background.
strange days have found us
strange days have tracked us down
they're going to destroy
our casual joys
we shall go on playing
or find a new town
--- Jim Morrison
We will of course revisit these concepts, but we have enough tools now to begin our examination
of the cockpit doors.
of the cockpit doors.
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