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In this section I’ll introduce the idea that a chaotic dynamical system
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like the logistic equation what r equals 4
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is a deterministic sources of randomness.
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In order to do so we’ll have to think carefully about what randomness means.
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What does it mean when we say a process or an outcome is random?
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I’ll build up a series of arguments layer by layer.
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None of these arguments are particularly technical
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in the sense of they don’t require calculate or algebra.
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However they are conceptually rich and a bit abstract.
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But I think we’ll I end up with some really interesting conclusions
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that will be perhaps surprising and I hope a lot of fun think about.
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So let’s get start it.
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I’ll start by introducing a technique known as symbolic dynamics.
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The idea behind symbolic dynamics is to convert an orbit, a series of numbers,
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in this case between 0 and 1, into a sequence of symbols.
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and the standard way to do this is as follows,
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if our iterate x is less than 0.5 , I’ll call that L
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and if x is greater than or maybe equal to 0.5, I’ll call that R
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so I am picturing this would be on the left half of the unit interval
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and this is on the right half.
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The symbols you use are completely arbitrary
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You can use hearts and spades or x and y or zeros and ones.
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But I’ll use L and R.
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So for example, suppose we had the following itinerary
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Ok, so here are the first couple iterates for the logistic equation
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Again r equals 4, and initial condition is 0.613
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So let’s convert this into symbolic dynamics
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So 0.613 that’s greater than a half that would be an R
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0.949 that’s also greater than a half that would be R
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0.194 is less than a half, less than .5, I call then L
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0.625 that’s greater than a half, that’s an R
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This is also greater than a half, so that would be an R
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So the idea is that I can take any itinerary,
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any orbit a sequence of number between 0 and 1
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and convert that into a sequence of symbols R R L R R in this case.
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So once we have symbols sequence, the idea is that we can study
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the dynamics of symbol sequence
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instead of the dynamics of the original orbit.
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And in many cases, one can show that properties of the orbit
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are the same as the properties of the symbol sequence.
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So studying the symbol sequence is just as good as the original orbit.
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So let me write that
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So properties are the same for the orbit and the symbol sequence.
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So when I say properties what I mean is
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is that say the existence of fixed points and
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the stability of fixed points
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the symbolical dynamical system involving
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just the symbols L and R would have
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the same number of fixed points
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and their stability would be the same
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and if the symbol sequence, the symbolic dynamical system
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has say sensitive dependence on initial conditions or aperiodicity
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then the original orbit, the original dynamical system would as well.
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Now this isn’t an obvious statement at all.
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Because it seems like by going to symbols
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I am throwing out a lot of information.
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After all any number that was between 0 and a half,
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I decided to just turn into L
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So that’s a very coarse thing to do.
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There are lots of lots of numbers and
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infinite number of numbers between 0 and a half.
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And I just turned all of those into L
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So it seems as if I am losing information
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so how can these two things be the same.
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Well, it turns out that for this particular way of forming symbols
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one can show and argue the following
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so let me do this with an example of sorts
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suppose I show you a symbol sequence R R L R L L R
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so then I might ask you what initial conditions could
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have given rise to this particular symbol sequence.
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And one can show you can kind of infer backwards
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to that would correspond to pretty narrow region of initial conditions
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and moreover that would just be a single connected region,
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that would give rise to this
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Then I could say, well, ok, what if the sequence was this
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and then you could show that the possible initial conditions
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that would have given rise to an orbit
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whose symbol sequence is this would be smaller still.
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and if I had another symbol, the possible initial conditions
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that give rise to this is smaller still
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And so in the limit that the symbol sequence becomes infinitely long.
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The possible initial conditions that would give rise to it become infinitely small.
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Another way to say this is that if you give me one single initial condition
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the symbol sequence that results from that is unique.
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there is one and only one symbol sequence that
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that one that results from that one single initial condition
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and that sort of make sense this is a deterministic dynamical system.
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So the key feature here is that
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there is one to one relationship between initial conditions and symbol sequences.
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So if you tell me the infinitely long symbol sequences
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I could know, I would know the initial condition
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And if I know the initial condition of the deterministic dynamical system
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that contains all the information about the orbit.
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So the infinite sequence encodes for the initial condition
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and the initial condition together with the dynamic tells you the orbit
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and from that one can get the properties.
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So I guess what I am trying to say is that the information in the symbol sequences
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is the same as information in the initial conditions.
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And ways of forming symbols from numbers
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that have this property are called
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generating in the particular scheme is sometimes called generating partition.
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So I don’t want to write down a formal definition of this.
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Because I think it will get us too far a field
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and get us into some really difficult notation
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but a partition and a partition was just to go back here
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this in a sense would be that a partition the description of the symbolic dynamics
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this tells me how to go from the orbit the x’s to the symbols the L and R
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that this scheme would be called the generating partition
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if longer and longer sequences encode for smaller and smaller
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and unique non-overlapping regions of initial conditions.
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Ok, so not all symbolic encoding schemes have this nice property
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So in fact if I had chose .4 as the cutoff
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so if x was less than .4 I call it L
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and it’s R otherwise
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then that would not have this property
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so it’s only special partition special ways of encoding that have this nice feature
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but the one that I described does indeed have this nice feature
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so it’s only one this is the case that this is true
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so let me just to make things little more accurate
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say, you know, if we use a generating partition
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so provided that we have a generating partition which we do in this case
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the properties of the orbit and the properties of symbol sequences
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are in the sense I’ve described the same.
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Lastly I want to mention that this technique of symbolic dynamics
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is a way of proving things about dynamical systems
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so I said in the last set of the lectures that it’s proved rigorously that
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when r equals 4 the logistic equation has sensitive dependence
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on initial conditions and the orbits are aperiodic.
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The way one would go about doing that proof
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and this is just a very very rough sketch
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would be to do this mapping from the dynamical, original dynamical system
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to symbol sequences proved properties of these symbol sequences
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and then if all of this holds which it would
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in this case what you prove about the symbol sequences
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which is which are easier to work with
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turn out to be true about the orbit as well.
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In any event now that we have this idea symbolic dynamics
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Let’s take a look at what symbolic dynamics
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look like for the logistic equation with r equals 4.