Sequences are complicated.
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It's not so easy to find the limit of a sequence.
The logistic map provides one source of rich, and by rich I mean very complicated.
Examples. Let's see how these are defined.
Here's a sequence I want to consider.
It's actually a family of sequences, which depend on some parameter r.
Let's call r, 2.5 for the time being. And I'll pick an initial
value for my sequence. Let's make it 0.25 for now.
And then here's the recursive definition of the sequence.
A sub n plus one will be this constant r times the
previous term a sub n times one minus a sub n.
And by varying r and by varying the initial value
I'll get a whole bunch of different sequences to consider.
Let's try it with those values. Let's try it with r equals 2.5.
So here's an example.
I start with the initial value, 0.25 and each subsequent value
is 2.5 times the previous value times one minus the previous value.
We have graphed these values, right?
The first is at 0.25, then it goes up
a bit, then it starts sort of levelling off here.
And indeed, if we look at this numerically, and
if the first value is exactly 0.25, and then as I run down through the
sequence, it seems like the values are getting closer and closer to about 0.6.
And in this case the limit of the sequence is in fact 0.6.
What if we start with a different initial value?
What if instead of a sub one equals 0.25, we make a sub one equal something else.
So the recursive formula we're using is a sub n plus one is
2.5 times a sub n times one minus a sub n. And
before, we were starting with a sub one equal to 0.25.
Let's start with 0.8 and see what happens in that case.
So if I start with 0.8, I could rewrite 0.8 as 4
5ths, and instead of writing 2.5 I'll write five
halves times a n times one minus a n.
Just like working with the fractions instead of the decimals.
So let's use this to calculate the next term.
What a sub two?
Well, according to this formula, it's five halves times
a sub one, times one minus a sub one.
But I've got a sub one, it's 4 5ths.
So that's five halves times 4 5ths minus one minus 4 5ths, which is 1 5th.
Now, 1 5th and this five can cancel. I can get rid of these.
And this four and this two, become a two in the numerator, so I've just got 2 5ths.
So a sub two is 2 5ths. Now I can calculate a sub three.
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So a sub three will be five halves again, times a sub
two, times one minus a sub two. And, in this case.
What is, a sub two, we just calculated, it's 2 5ths.
So this is five halves times 2 5ths times one minus 2
5ths, which is 3 5ths. But now I've got five halves and
2 5ths, and those cancel, so a sub three is just 3
5ths. Now, let's calculate a sub four.
Well, a sub four is, again, five halves times
a sub three times one minus a sub three.
That's five halves times a sub three is 3 5ths.
One minus 3 5ths is 2 5ths.
But now I've
got 5 halves and 2 5ths. Those cancel.
So a sub four is just 3 5ths. Or what's a sub five?
Well, a sub 5 is also 3 5ths.
I'm going to use the exact same formula here but I'm again just
going to plug in 3 5ths and I'm going to get 3 5ths out.
So a sub five is 3 5ths a sub six is 3 5ths, right.
The point is that this sequence is just constant from
here on out. And that means that, in this case the
limit as n approaches infinity of the sequence is just 3 5ths,
just like in the case when I started with 0.25, alright.
It happened in this case that even when I start with 0.8,
when I use this formula I again ended up with the same limit.
It's kind of interesting. Let's try
r equals 7 3rds.
I'll again, have my sequence start with 0.25, but then each new term
is 7 3rds times the old term, times one minus the old term.
And if we look at this graph, it looks like the
sequence is converging, and if we look at the numbers, right.
The first term in this sequence is 0.25 and it's
about 0.4 and about 0.57 and it keeps on going and
it does indeed look like, the sequence is getting closer and closer to something.
This number might not be too meaningful to you, but you can
in fact show that the limit of this sequences is 4 7ths.
And indeed, this number is about 4 7ths.
So changing the value of r seems to affect the limit.
Let's try r equals 3.25.
So again my sequence will start
with 0.25, but now the r value is 3.25. What does the sequence look like?
Well, I can graph a bunch of terms of the sequence, and it starts with
a 0.25 and it goes up, and then it seems to bounce between two values.
And indeed, the numerical evidence supports that same conclusion.
Here's the first value, 0.25 and it's about 0.6,
0.7, 0.5, 0.8, 0.5, 0.8, 0.5.
And since it's been flip-flopping between these two values.
And since it's flip-flopping between those two values, the limit doesn't exist.
Let's try another example.
Let's try r equals 3.7.
The same initial value 0.25, but now 3.7 is my value for r.
What does this sequence look like? Well,
I can start graphing the terms in this sequence and wow.
I mean, it just looks like garbage.
There just doesn't seem to be any pattern at all, you know?
And there's maybe moments when it looks like things
are getting better, but then it suddenly breaks apart again.
And if you look at the numbers, right.
Numerically things don't look so great here either.
I mean 0.25, that's the initial value but as you look through
these numbers, it doesn't look like any sort of pattern.
These really coming out.
What is going on here?
Well changing the value of r doesn't
just change quantitative features of the sequence.
It changes qualitative features of the sequence.
Depending on the value of r, this sequence might converge, might have a limit.
It might flip-flop between a couple
values, might flip-flop between four different
values, it might just move all over the place
and not really have any kind of discernible pattern.
And it all boils down to this value of r in a seemingly mysterious way.
But that's not to say that the sequence
can't be understood, that, that it can't be studied.
Alright.
Maths isn't just random, right, I mean there's structure to this thing,
and with more work you can really start digging in, to the very
complicated structure that appears, even something as
seemingly simple as just the sequence of numbers.
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