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