Since this is the first homework solution 
in this course I want to

take a little bit of time 
to explain the symbols.

The green circles.

Everyone should do,

and there will be solutions in these 
videos everyone should try

all of the problems with the blue squares.

Those problems will be discussed 
on the forum,

the black diamonds are for people 
who know a little bit more,

who won a challenge.

These are completely optional and 
again will be discussed on the

forum,

the first problem here will call 
upon the logistic map program

that you were in unit 1.2.

Here's mine that led version,

which I showed you in the solution 
to that quit the task in this

problem was to generate a trajectory 
from Mexico also point to 200

points lower with articles 2 and 
then to generate a trajectory,

using the same are in the 
same number of points,

but a slightly different initial 
condition the next task was to

plot the absolute value of the 
difference between those

trajectories versus and here's the 
difference here I'm generating

the end is setting up a 
figure doing the plot.

There is the figure.

Let's keep that in mind,

then we were supposed to repeat this 
for article 3.4 and our equal

3.7-2.

So I'll do that quickly here is that 
figure asked ask workers to

do the same thing for equals 3.7.

2,

there is the 3rd party the next task 
was to compare these parts to

the ones in the homework.

And then answer the following questions.

So which of these clots can corresponds 
to Oracle's too.

That was the one that fell 
off like a stone.

That was.

See which other parts and figure 
one corresponds to Oracle's 3.4

that was the one but isolated 
as it was falling.

That was it.

And then a 3rd question which 
of the plots and figure one

corresponds to the plot that we 
generate with our equals 3.7.

2,

that was the chaotic one in B now.

The point of this problem goes back 
to my example of an Eddie in a

stream as a metaphor for chaos and 
the notion of dropping 2 wood

chips and that any very close together 
and watching how fast they

separate if the air tractor is a 
fixed point and you drop those 2

wood chips those 2 initial conditions 
in the basement of

attraction of that a tractor.

Both of those initial conditions 
would be converging to the fix

pointed tractor.

So the absolute value of the 
difference between then,

which is the distance between them 
would converge to zero if their

tractor or a periodic orbit.

Then the 2 initial conditions would 
rattle in from 2 different

directions.

So the distance between them might 
isolate but eventually they

would end up on the same periodic 
orbit and so that distance would

converge to a fixed value not necessarily 
zero because they might

be on different points 
on the periodic orbit.

Kind of like 2 cars going 
around a racetrack,

they might be going slower in the 
corners and faster on the street

stretches so they wouldn't just stay 
directly opposite each other

all the time,

although as you can see in this case 
the difference does converged

to zero if their tractor is chaotic 
if the attractiveness chaotic

the 2 initial conditions will move 
chaotic they through the air

tractor and the distance between them,

we'll also changed periodically.

That's a problem,

you need to generate 2 trajectories 
500 points along with slightly

different national conditions and 
look at the last number in each

of those trajectories there at the 
2 trajectories looks pretty

small.

To me the choice isn't in the problem 
worth 0.2-5 1 and none of

the above 10 to the minus 17th.

It's pretty darn close to zero.

So I would select the first answer 
in part of the idea was to

repeat that for article 3.4.

There's the calculation.

And again,

it looks like the answer is zero.

Park Ji of this problem required 
a little bit more programing.

Here's a 5,000 point trajectory 
at our equals 3.7-2.

Here's a 5,000 point trajectory from 
a slightly different initial

condition at the same our value.

Here's a vector containing an element 
was difference of those 2

trajectories and with the absolute 
value taken and here's the

average of the values in that vector 
Nazi which of the answers

that corresponds to looks like that 
one the next problem was about

extending that calculation.

After 500,000.

The answer doesn't change a whole lot,

but it is a little different,

it's 2.4-4 one,

the fact that that difference doesn't 
change very much between

5,000 and 500,000 points is pretty 
amazing what that says is that

as the initial conditions move 
around the chaotic a tractor.

The average distance between them 
is pretty much the same over

5,000 points were over 500,000 points 
that's a consequence of the

combination of sensitive dependence 
on initial conditions and the

bounded patterned structured nature 
of a chaotic attractive in

problem too.

We went back to using the app,

the task was to generate a 50.0 trajectory 
from ex not equals 0.2

using this very carefully chosen 
our primary value let's restart

the simulation this doesn't look 
to me like anything periodic or

anything that's a fixed point,

I would guess this is chaotic,

but there's something very 
interesting going on here.

Look at the sort of peace and 
then the sort of peace.

There's some patterns going on there,

but they're not quite the same.

So I would guess that this is a 
chaotic orbit in Part B of this

problem.

We're gonna watch for a little bit 
longer and see what happens to

see if it's really chaotic or 
if it's gonna settle down to

something with this out.

You can watch an ongoing process 
of iterations by clicking Start

animation.

I still see that little Arrowhead 
thing coming through and then

things look like they've go chaotic 
in between the work it

re-occurrence of that pattern 
makes me suspect were nearby

objectification point.

Now things are gone,

periodic and I'm gonna stop the animation,

so we can count and see 
what kind of period.

It is,

looks like it repeats every 123456789.

That's a 9 cycle,