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,