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Silent introduction
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Let's step back for a minute and take stock of the main message of what we've seen so far.
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The purpose of looking at the logistic map
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was to illustrate the phenomenon of chaos,
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which means seemingly random behavior
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with sensitive dependence on initial conditions.
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The logistic map was a simple, completely deterministic equation
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that, when iterated, can display this kind of seemingly random
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behavior with sensitive dependence (depending on the value of R).
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There was no randomness in the logistic map equation
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Completely determined what the next value of x will be
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by the previous value of x
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And yet we see this seemingly random behavior
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This is called deterministic chaos, where chaos arises from a
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completely deterministic system or equation.
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And the message to be learned here is that, if we have
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deterministic chaos, perfect prediction, a la Laplace's deterministic
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clockwork universe, which we saw earlier, is impossible,
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even in principle, since we can't know
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the precise value of the initial condition.
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This is a profound negative result that, along with quantum mechanics, helped wipe out the optimistic 19th
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century view of the clockwork Newtonian universe that kept going along in its predictable path in a deterministic way.
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But what's the positive message?
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Dynamical systems theory is the attempt to discover general principles concerning systems that change over time.
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Through the work of many people and studying logistic map
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and other similar deterministic equations, the result was to find some equally surprising profound and
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positive results which is what we're going to call universality in chaos.
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That is, characteristics that are universal across a certain broad set of chaotic systems.
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In short, while chaotic systems are not predictable in detail,
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there is a wide class of chaotic systems that have highly predictable universal properties.
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Let's look at what these are.
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Recall our logistic map bifurcation diagram,
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which showed the period doubling route to chaos
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And we had different, what we call, regimes of attractors. Remember we had our fixed point attractors?
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And we had our periodic attractor regime.
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And this led to a region of chaos where we have
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what's called a chaotic attractor -- or what's also known as a strange attractor.
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Note that there's some interesting looking structure in this chaotic attractor regime,
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including places where the system goes back to being periodic.
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Talking about that is really interesting, but it's beyond the scope of this particular course.
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But there's something that is really surprising and interesting
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and that is that this kind of period doubling route to chaos
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that we see here also appears in many other chaotic systems.
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The kinds of systems that "periodic route to doubling" appears in are what are called
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the unimodal or one-humped maps.
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Just like our logistic map that we're looking at right now,
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you can see that there's a single hump in the parabola
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And there are many such systems in nature that have this kind
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of behaviour. They all have the same period doubling route
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to chaos. Another map that shows that unimodal behaviour
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is the so-called sine map.
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It's called the sine map because it uses the trigonometric function of sine.
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Here's what it looks like.
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x sub t plus 1 = R/4 sin of (pi x sub t).
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Now if you've forgotten trigonometry, don't worry.
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Just think of this as a particular mathematical function that can
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be iterated in the same way that we iterated the logistic map.
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And notice here that the x doesn't stand for population anymore.
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We're now in the realm of abstract mathematical functions
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where x can stand for anything as long as it's between 0 and 1.
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People often write this R/4 term as the Greek letter lambda.
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I've left it as R over 4 just to keep the connection with the
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previous notion of R in the logistic map.
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But just think of this as a particular function
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that we're going to iterate.
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I've implemented the sine map as a NetLogo model
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which you can download from the course materials website
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And you can see that it also forms this parabola,
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or one-hump map, as a logistic function
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and it also displays the characteristic
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period doubling route to chaos.
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In the course materials site, it's called sinemap.nlogo
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So now we've seen one universal property of chaos
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in unimodal maps -- that is, the period doubling route to chaos
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But there's another, even more surprising, universality that
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was discovered in the 1970s and 1980s
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by a few different groups of people.
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I'll explain it by looking again at the logistic map bifurcation
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diagram. Now in the 1970s, the physicist Mitchel Feigenbaum,
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studied this map in great depth.
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And he measured, as precisely as he could, the points
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at which these various bifurcations occur.
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And he found that the period two bifurcation appears
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at about R = 3.0, period four at this value
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period eight at this value
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period sixteen at this value and so on
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until finally, at a value of approximately 3.569946 followed by
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some other decimal places, we have
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a period of infinity. That is, we have chaos. It's aperiodic.
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And that point is called "the onset of chaos."
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Now if you notice, these bifurcations are coming quicker
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and quicker as we vary R. So this goes from about .24 to .30
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That's a long time. And then from .30 to .34 something,
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that's shorter. And then shorter & then shorter & then shorter.
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So what Feigenbaum did was he took these bifurcations
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and he measured the rate at which the distance between
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bifurcations is shrinking. The rate at which these are shrinking is just defined as the ratio of
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this length
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divided by the length to the next bifurcation
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and then the ratio of this length
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from the period two bifurcation to the period four bifurcation
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divided by the length of the time from the period 4 bifurcation
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to the period 8 bifurcation and so on.
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Each of those ratios is a measure of the rate at which
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the bifurcations are shrinking.
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Let's take a look at these ratios mathematically.
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This is what they look like.
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The first ratio is the distance between the period 2 and
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the period 4 bifurcations. That's R2 - R1
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divided by the distance between the period 4 and the period 8 bifurcations.
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Ok that gives us 4.7514 etc and that's our first estimate of
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the rate at which the distance between the bifurcations is shrinking.
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Ok. Then we can go to the next set and get the distance
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between the period 4, period 8 divided by period 8, period 16.
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That gives us 4.656 etc. And we keep doing that
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And Feigenbaum did this using just a desktop calculator, and later a faster computer,
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and found that, as we keep doing this -- as these Rs get bigger and bigger and bigger -- this number starts
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converging to this value -- 4. 6692016
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followed by some decimal places.
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And in mathematical terms, that's the limit as n approaches infinity
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of R n+1 minus R of n divided by R n+2 minus R of n+1.
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But if you're not familiar with limits, don't worry about that.
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Just look at these examples and you can see that,
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as we increase the bifurcation number, we're going to get closer and closer to this value.
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In other words, each new bifurcation appears about
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this many times faster than the previous one.
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That's what Feigenbaum found.
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And this number has now become called Feigenbaum's constant
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because, amazingly, not only did he derive this mathematically after he observed it
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(he developed a whole theory that would show why this had to be true mathematically)
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he also showed that any unimodal or one-humped map
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like the logistic map or the sine map
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will have the same value for this rate.
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So this number is the universal constant for chaotic systems with one humped maps.
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This number has also been experimentally observed
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in a number of systems ranging from fluid flow to electric circuits, lasers, chemical reactions and so on.
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Another amazing fact about this is that
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at almost exactly the same time that Feigenbaum was carrying out his studies, the same thing was done by
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another research team -- the French mathematicians Pierre Collet and Charles Tresser
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They were completely independent of Feigenbaum
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And this co-discovery is now sometimes called
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The Feigenbaum, Collet, Tresser Theory.
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In summary, let's look at what we've learned in this unit as to the significance of dynamics in chaos for complex systems.
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Perhaps most importantly, we've seen an example
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in which complex, unpredictable behavior arises from
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very simple deterministic rules, such as the logistic map.
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We've also seen that dynamical systems theory gives us a
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vocabulary for describing complex behaviour using terms such as
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"attractors" and "period doubling route to chaos" and so on.
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We've seen that there are fundamental limits to detailed
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prediction in systems with chaos due to "sensitive dependence on initial conditions."
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At the same time, we've seen that there are universal properties of chaotic systems
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and we might call this "order in chaos."