WEBVTT
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Today's lecture is going to be
basically devoted to working out
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a single example of a nonlinear
system, but it is a very good
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example because it illustrates
three things which you really
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have to know about nonlinear
systems.
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I have indicated them by three
cryptic words on the board,
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but you will see at different
points in the lecture what they
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refer to.
Each of these represents
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something you need to know about
nonlinear systems to be able to
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effectively analyze them.
In addition,
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I am not allowed to collect any
more work, according to the
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faculty rules,
after today.
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And, of course,
I won't.
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But, nonetheless,
the final will contain material
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from the reading assignment G7,
and I will be touching on all
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of these today,
and 7.4, just a couple of pages
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of that.
You will see its connection.
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It discusses the same example
we are going to do today.
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And then I suggest those three
exercises to try to solidify
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what the lecture is about.
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Instead of introducing the
nonlinear system right away that
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we are going to talk about,
I would like to first explain,
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so that it won't interrupt the
presentation later,
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what I mean by a conversion to
a first-order equation.
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For that why don't we look at --
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Maybe I can do it here.
Our system looks like this,
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dx over dt.
For once, I am not writing x
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prime because I want to
explicitly indicate what the
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independent variable is.
It is a nonlinear autonomous
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system, so there is no t on the
right-hand side.
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But this is not a simple linear
function, ax plus by.
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It is more complicated,
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like you had in the
predator-prey robin-earthworm
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problem that you worked on last
night.
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And the other equation will be
g of (x, y).
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This is the system,
and explicitly it is nonlinear
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in general and autonomous,
as I have indicated on the
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right-hand side.
Now, remember that the
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geometric picture of this was as
a velocity field.
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You made a velocity field out
of the vectors whose components
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were f and g,
so that is (f)i plus (g)j,
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but what it looked like was a
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plane filled up with a lot of
vectors pointing different ways
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according to the velocity at the
point was.
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And side-by-side with that went
the solutions.
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A typical solution was x equals
x of t,
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y equals y of t
represented as a column vector.
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And that is a parametric
equation.
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And the geometric picture of
that was given by its
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trajectory.
When you plotted it,
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it was a trajectory.
It had not only the right
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direction at each point,
in other words,
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but it also had to have the
right velocity at each point.
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In other words,
sometimes the point was moving
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rapidly and sometimes it was
moving more slowly along that
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path.
And the way it moved was an
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important part of the solution.
Now, what I plan to do is --
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The conversation that I am
talking about takes place by
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eliminating t.
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Now, why would one want to do
that?
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t, after all,
is an essential part of the
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solution.
It is an essential part of this
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picture because without t you
would not know how long the
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arrows were supposed to be.
You would still know their
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direction.
And, of course,
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it occurs in here.
Now, let's take the first step
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and eliminate t from the system
itself.
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As you see, that is very easy
to do because the right-hand
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side has no t in it anyway and
the left-hand side has the t
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only as the denominators of the
differential quotients.
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The obvious way to eliminate t
is just to divide one equation
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by the other.
And since usually we think of
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not dx by dy,
but dy by dx,
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I will vow --
What is dy over dx?
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Well, if I divide this equation
by that equation,
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dy by dt divided by dx by dt,
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according to the chain rule,
or according to commonsense,
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you cancel the dt's and you get
dy over dx.
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And, on the right-hand side,
you get g of (x,
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y) divided by f of (x,
y).
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But what is that?
That is the equation of the
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first day of the term.
By taking that single step,
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I convert this nonlinear
system, which we virtually never
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find explicit solutions to,
into an ordinary first order
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equation which,
in fact, you also don't usually
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find explicit solutions to,
except with our narrow blinders
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on.
We only get problems at the
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beginning of 18.03 where there
will be an explicit solution to
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this.
I have converted the nonlinear
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system to a single first order
equation to which I can apply
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the usual first order methods
that include hope that I will be
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able to solve it.
Now, what do I lose?
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Well, what happens to this
picture?
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If you don't have the t in it
anymore then you don't have
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velocity, you don't have arrows
with a length to them because
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the length gives you how fast
the point is going.
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Without t I don't know at any
point how fast it is supposed to
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be going.
All I know at any point (x,
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y) is no longer dx over dt
and dy over dt.
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All I know is dy over dx.
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In other words,
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the corresponding picture,
if I eliminate t from this
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picture all that is left is the
directions of each of these
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arrows.
What disappears is their
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length.
And, in fact,
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since dy over dx is
purely a slope,
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I cannot even tell whether the
point is traveling this way or
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that way.
It doesn't make any sense to
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have the point traveling anymore
since there is no time in which
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it can do its traveling.
So that picture gets changed to
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the direction field.
The corresponding picture now,
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all you do is take each of
these arrows,
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you snip it off to a standard
length, being careful to snip
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off the little pointy end,
and it becomes nothing but a
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line element.
And all it has is a slope.
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What corresponds to the
velocity field here is now just
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our slope field or our direction
field.
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All we know is the slope at
each point.
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How about the solution?
Well, if I have eliminated
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time, the solution is no longer
a pair of parametric equations.
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It is just an equation
involving x and y.
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I can hope that the solution
will be explicit,
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y equals y of x,
but you already know from the
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first days of the term that
sometimes it isn't.
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Let's call that explicit
solution.
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Often you have to settle for
it.
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And sometimes it is not a pain.
It is something that is even
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better.
Settle for a solution which
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looks like this which is y
defined implicitly as a function
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of x.
And what does the picture of
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that look like?
Well, the picture of this is
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now simply an integral curve.
It is not a trajectory anymore.
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It is what we called an
integral curve.
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I am reminding you of the very
first day of the term,
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or maybe the second day.
And all it has at each point
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the right slope because that is
all the field is telling me now.
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It only has a slope at each
point.
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It doesn't have any magnitude
or direction.
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It has the direction,
but it doesn't have direction
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in the sense of an arrow telling
me whether it is going this way
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or the opposite way.
That is the picture with t in
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it.
These are the corresponding
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degeneration of that.
It is a coarsening,
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it's a cheapening of it.
It is throwing away
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information.
Throw away all the information
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that had to do with t,
and we are back in the first
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days of the terms with a single
first order equations involving
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only x and y with solutions that
involve only x and y with curves
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that are the graphs of these
solutions but again have no t in
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them.
In effect, they are just the
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paths of the trajectories.
Whereas, the trajectory is the
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point that shows actually how
the point is moving along in
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time faster or slower at various
points.
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Now, why would one want to lose
information?
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Well, the great gain is that
this might be solvable.
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Whereas, this almost certainly
is not.
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That is a big plus.
For example,
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let's illustrate it on the very
simplest possible case.
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I cannot illustrate it on a
nonlinear system very well,
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or not right now because,
in general, nonlinear systems
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are not solvable.
Let's take an easy example.
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Suppose the system were a
linear one, let's say this one
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that we have talked about
before, in fact.
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That is a simple linear
two-by-two system.
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This means the derivatives with
respect to time.
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Well, you know that the
solutions we talked about in
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fact, last time,
typical solutions that would be
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something like (x,
y) equals a constant times,
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let's say if x is cosine t
then y would be the
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derivative of that,
which is negative sine t.
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Or, another one would be c2
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times sine t,
cosine t.
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And those, of course,
are circles.
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They are parametrized circles,
so they are circles that go
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around, in fact,
in this direction.
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And they have a certain
velocity to them.
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Now what do I do if I follow
this plan of eliminating t?
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Well, if I eliminate t directly
from the system,
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what will I get?
I will get dy by dx.
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I divide this equation by that
one.
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-- is equal to negative x over.
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Oh, well, of course,
that is solvable.
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You were able to solve by 18.01
methods before you ever come
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into this course.
By separation of variables,
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it is y dy equals
negative x dx,
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which integrates to be one-half
y squared equals negative
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one-half x squared plus a
constant.
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And after multiplying through
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by two and moving things around
it becomes x squared plus y
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squared.
The nicest way to say it is
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implicitly, x squared plus y
squared equals some positive
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constant.
These are the circles.
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Now, can I eliminate t?
I could also take one of these
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solutions and eliminate t.
If I square this,
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the best way to do it is not to
use our cosines and arcsines and
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whatnot, which will just get you
totally lost.
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Square this,
square that and add them
00:13:06.000 --> 00:13:12.000
together.
And you conclude that x squared
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plus y squared is equal to one.
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Or, if c1 is not one it is
equal to c1 squared.
00:13:16.000 --> 00:13:22.000
You could eliminate t from the
solution the way you eliminate t
00:13:21.000 --> 00:13:27.000
from a pair of parametric
equations, and you get it the
00:13:25.000 --> 00:13:31.000
same way that x squared plus y
squared, in this case,
00:13:29.000 --> 00:13:35.000
would be c1 squared, I guess.
00:13:39.000 --> 00:13:45.000
In some sense,
what we are doing is cycling
00:13:42.000 --> 00:13:48.000
around to the beginning of the
term.
00:13:44.000 --> 00:13:50.000
In fact, this whole lecture,
as you will see,
00:13:48.000 --> 00:13:54.000
is about cycles of one sort or
another.
00:13:51.000 --> 00:13:57.000
But I keep thinking of that
great line of poetry,
00:13:54.000 --> 00:14:00.000
"in my end is my beginning" or
maybe it's "in my beginning is
00:13:59.000 --> 00:14:05.000
my end".
I think both lines occur.
00:14:03.000 --> 00:14:09.000
But that is what is happening
here.
00:14:05.000 --> 00:14:11.000
This is the beginning of the
course and this is the end of
00:14:08.000 --> 00:14:14.000
the course, and they have this
almost trivial relation between
00:14:12.000 --> 00:14:18.000
them.
But notice the totally
00:14:14.000 --> 00:14:20.000
different methods used for
analyzing this,
00:14:17.000 --> 00:14:23.000
where the t is included,
than from analyzing this where
00:14:20.000 --> 00:14:26.000
there is no t.
The goals are different,
00:14:22.000 --> 00:14:28.000
what you look for are
different, everything is
00:14:25.000 --> 00:14:31.000
different, and yet it is almost
the same problem.
00:14:30.000 --> 00:14:36.000
I promised you a nonlinear
example.
00:14:33.000 --> 00:14:39.000
I guess it is time to see what
that is.
00:14:44.000 --> 00:14:50.000
It is going to be another
predator-prey equation,
00:14:47.000 --> 00:14:53.000
but one which is,
in some ways,
00:14:49.000 --> 00:14:55.000
simpler than the one I gave you
for homework.
00:14:52.000 --> 00:14:58.000
The predator is going to be x
because you are used to that
00:14:56.000 --> 00:15:02.000
from the robins.
I am keeping the same
00:14:59.000 --> 00:15:05.000
predator-prey variables.
And the prey will be y.
00:15:03.000 --> 00:15:09.000
And now just notice the small
difference from what I gave you
00:15:08.000 --> 00:15:14.000
before.
For one thing,
00:15:09.000 --> 00:15:15.000
I am not giving you specific
numbers.
00:15:12.000 --> 00:15:18.000
I am going to,
at the beginning at least,
00:15:15.000 --> 00:15:21.000
do some of the analysis at the
beginning using letters,
00:15:19.000 --> 00:15:25.000
using parameters.
a, b, c, d are always going to
00:15:22.000 --> 00:15:28.000
be positive constants.
I am going to assume that x
00:15:26.000 --> 00:15:32.000
prime equals minus ax.
00:15:30.000 --> 00:15:36.000
This represents the predator
dying out if there is no prey
00:15:36.000 --> 00:15:42.000
there, but if there is something
to eat that term represents the
00:15:43.000 --> 00:15:49.000
predator meeting up with the
prey and gobbling it up.
00:15:48.000 --> 00:15:54.000
And b is the coefficient.
How about without predictors,
00:15:54.000 --> 00:16:00.000
the prey will multiply and be
fruitful.
00:16:00.000 --> 00:16:06.000
Unfortunately they get eaten,
and so there will be a term
00:16:04.000 --> 00:16:10.000
that looks like this.
Now, there are two basic
00:16:07.000 --> 00:16:13.000
differences between these simple
equations and the slightly more
00:16:12.000 --> 00:16:18.000
complicated ones you had with
the robin-earthworm equations.
00:16:16.000 --> 00:16:22.000
Namely, in the first place,
I am assuming that these guys,
00:16:20.000 --> 00:16:26.000
let's give them names.
I want you to remember which is
00:16:24.000 --> 00:16:30.000
x and which is y,
so we are going to think of
00:16:27.000 --> 00:16:33.000
these are "sharx".
[LAUGHTER]
00:16:31.000 --> 00:16:37.000
And these will be food fish,
so let's make them "yumfish".
00:16:50.000 --> 00:16:56.000
The fact that with your robins
you had a positive term here.
00:16:54.000 --> 00:17:00.000
I made this 2x.
And now it is minus a times x.
00:16:57.000 --> 00:17:03.000
What is the difference?
The difference is that robins
00:17:02.000 --> 00:17:08.000
have other things to eat,
so even if there are not any
00:17:06.000 --> 00:17:12.000
worms, a robin will survive.
It could eat other insects,
00:17:10.000 --> 00:17:16.000
grubs, Japanese beetle grubs.
I think it can even eat seeds.
00:17:15.000 --> 00:17:21.000
Anyway, the robins in my garden
seem to be pecking at things
00:17:20.000 --> 00:17:26.000
that don't seem to be insects.
And the other is that I am
00:17:24.000 --> 00:17:30.000
assuming a very naīve growth
law.
00:17:27.000 --> 00:17:33.000
For example,
if there are no sharks,
00:17:30.000 --> 00:17:36.000
how are the food fish fruitful
and multiplying?
00:17:35.000 --> 00:17:41.000
They multiple exponentially.
Now, obviously you cannot have
00:17:39.000 --> 00:17:45.000
unlimited growth like that.
With the worms we added the
00:17:43.000 --> 00:17:49.000
term minus a constant times y
squared to indicate that even
00:17:48.000 --> 00:17:54.000
worms cannot multiply purely
exponentially forever,
00:17:51.000 --> 00:17:57.000
but ultimately their growth
levels off because they cannot
00:17:56.000 --> 00:18:02.000
find enough organic matter to
plow their way through.
00:18:01.000 --> 00:18:07.000
Those are the two differences.
I am not assuming a logistic
00:18:05.000 --> 00:18:11.000
growth law.
This is less sophisticated than
00:18:08.000 --> 00:18:14.000
the one I gave you for homework.
This assumes that sharks have
00:18:12.000 --> 00:18:18.000
absolutely nothing to eat except
these fish, which is not so bad.
00:18:17.000 --> 00:18:23.000
That's true,
more or less.
00:18:25.000 --> 00:18:31.000
My plan is now,
with this model,
00:18:27.000 --> 00:18:33.000
let's start the analysis,
as you learned to do it.
00:18:30.000 --> 00:18:36.000
And we are going to run into
trouble at various places.
00:18:34.000 --> 00:18:40.000
And the troubles will then
illustrate these three points
00:18:38.000 --> 00:18:44.000
that I wanted to add to your
repertoire of things to do with
00:18:43.000 --> 00:18:49.000
nonlinear systems when you run
into trouble.
00:18:47.000 --> 00:18:53.000
And it will also increase your
understanding of the nonlinear
00:18:50.000 --> 00:18:56.000
systems.
The first thing we have to do
00:18:52.000 --> 00:18:58.000
is find the critical points.
I am going to assume that you
00:18:56.000 --> 00:19:02.000
are good at this and,
therefore, not spend a lot of
00:18:59.000 --> 00:19:05.000
time detailing the calculations.
I will simply write them on the
00:19:04.000 --> 00:19:10.000
board.
The first equation I am going
00:19:06.000 --> 00:19:12.000
to write down is x times minus a
plus by equals zero.
00:19:11.000 --> 00:19:17.000
And I assume you know why I am
00:19:13.000 --> 00:19:19.000
writing that down.
And the other equation will be
00:19:17.000 --> 00:19:23.000
y times c minus dx is zero.
00:19:20.000 --> 00:19:26.000
Those are the simultaneous
equations I have to find to find
00:19:24.000 --> 00:19:30.000
the critical point.
The first one is if the product
00:19:30.000 --> 00:19:36.000
of these is zero,
either x is zero or the other
00:19:34.000 --> 00:19:40.000
factor is.
So, from the second equation,
00:19:38.000 --> 00:19:44.000
if x is zero,
y has to be zero also.
00:19:42.000 --> 00:19:48.000
That is one critical point.
Now, if x is not zero then this
00:19:48.000 --> 00:19:54.000
factor has to be zero which says
that y must be equal
00:19:54.000 --> 00:20:00.000
to a over b.
And if y is equal to a over b,
00:19:59.000 --> 00:20:05.000
it is not zero here.
Therefore, this factor must be
00:20:02.000 --> 00:20:08.000
zero which says that x is equal
to c over d.
00:20:06.000 --> 00:20:12.000
You see right away that this
must be a simpler system because
00:20:10.000 --> 00:20:16.000
it is only producing two
critical points.
00:20:12.000 --> 00:20:18.000
Whereas, the system that you
did for homework had four.
00:20:16.000 --> 00:20:22.000
Here is one.
Well, let's just write them up
00:20:19.000 --> 00:20:25.000
here.
The critical points are zero,
00:20:21.000 --> 00:20:27.000
zero.
That doesn't look terribly
00:20:23.000 --> 00:20:29.000
interesting, but the other one
looks more interesting.
00:20:27.000 --> 00:20:33.000
It is c over d,
a over b.
00:20:31.000 --> 00:20:37.000
Well, let's take a look first
at the zero, zero critical
00:20:34.000 --> 00:20:40.000
point.
The origin, in other words.
00:20:37.000 --> 00:20:43.000
What does that look like?
00:20:44.000 --> 00:20:50.000
Well, at the origin,
the linearization is extremely
00:20:47.000 --> 00:20:53.000
easy to do because I simply
ignore the quadratic terms,
00:20:51.000 --> 00:20:57.000
which are the product of two
small numbers,
00:20:55.000 --> 00:21:01.000
where these only have a single
small number in it.
00:20:58.000 --> 00:21:04.000
It is minus ax,
cy.
00:21:01.000 --> 00:21:07.000
In other words,
the linearization matrix is
00:21:03.000 --> 00:21:09.000
minus a, zero,
zero and cy gives me a
00:21:06.000 --> 00:21:12.000
coefficient c there.
00:21:08.000 --> 00:21:14.000
Now, I don't think at any point
I have ever explicitly told you
00:21:12.000 --> 00:21:18.000
that I hope you have learned
from the homework or maybe your
00:21:16.000 --> 00:21:22.000
recitation teacher,
but for heaven's sake,
00:21:19.000 --> 00:21:25.000
put this down in your little
books, if you have a diagonal
00:21:23.000 --> 00:21:29.000
matrix, for god's sake,
don't calculate its
00:21:26.000 --> 00:21:32.000
eigenvalues.
They are right in front of you.
00:21:30.000 --> 00:21:36.000
They are always the diagonal
elements.
00:21:32.000 --> 00:21:38.000
The eigenvalue,
you can check this out if you
00:21:35.000 --> 00:21:41.000
insist on writing the equation,
but trust me it is clear.
00:21:40.000 --> 00:21:46.000
If I, for example,
subtract c from the main
00:21:43.000 --> 00:21:49.000
diagonal, I am going to get a
determinant zero because the
00:21:47.000 --> 00:21:53.000
bottom row will be all zero.
The eigenvalues are negative a
00:21:51.000 --> 00:21:57.000
and c.
In other words,
00:21:53.000 --> 00:21:59.000
they have opposite signs.
This is a negative number.
00:21:56.000 --> 00:22:02.000
That is a positive number --
-- because a,
00:22:00.000 --> 00:22:06.000
b, c and d are always positive.
And, therefore,
00:22:03.000 --> 00:22:09.000
this is automatically a saddle.
You don't have to calculate
00:22:07.000 --> 00:22:13.000
anything.
It is all right in front of
00:22:09.000 --> 00:22:15.000
you.
It must be a saddle and,
00:22:11.000 --> 00:22:17.000
therefore, unstable because all
saddles are.
00:22:13.000 --> 00:22:19.000
And, in fact,
you can even draw the little
00:22:16.000 --> 00:22:22.000
picture of what the stuff looks
like near the origin without
00:22:19.000 --> 00:22:25.000
even bothering to calculate
eigenvectors,
00:22:22.000 --> 00:22:28.000
although it is extremely easy
to do.
00:22:24.000 --> 00:22:30.000
Just from common sense,
these are the sharks and these
00:22:27.000 --> 00:22:33.000
are the yumfish.
Well, if there are zero
00:22:31.000 --> 00:22:37.000
yumfish, in other words,
if I am on the sharks axis,
00:22:35.000 --> 00:22:41.000
the axis of sharks,
I die out.
00:22:38.000 --> 00:22:44.000
Well, forget about this side.
It's on the positive side.
00:22:42.000 --> 00:22:48.000
That makes sense.
But I die out because the
00:22:45.000 --> 00:22:51.000
sharks have nothing to eat.
Whereas, if I am on the yumfish
00:22:50.000 --> 00:22:56.000
axis I go this way.
I grow because,
00:22:52.000 --> 00:22:58.000
without any sharks to eat them,
the yumfish increase.
00:22:56.000 --> 00:23:02.000
So it must look like that.
And, therefore,
00:23:00.000 --> 00:23:06.000
the saddle must look like this.
The saddle curves hug those and
00:23:04.000 --> 00:23:10.000
go nearby.
Now, the other critical point
00:23:07.000 --> 00:23:13.000
is the interesting one.
And for that the analysis,
00:23:10.000 --> 00:23:16.000
in order that you don't spend
the rest of this period writing
00:23:14.000 --> 00:23:20.000
a's, b's, c's and d's,
I am going to make the
00:23:17.000 --> 00:23:23.000
simplifying assumption.
But it doesn't change
00:23:20.000 --> 00:23:26.000
qualitatively any of the
mathematics.
00:23:22.000 --> 00:23:28.000
It just makes it a little
easier to write everything out.
00:23:26.000 --> 00:23:32.000
I am going to assume that
everything is one.
00:23:30.000 --> 00:23:36.000
Well, in fact,
even this would be good enough,
00:23:33.000 --> 00:23:39.000
but let's make everything one.
I am going to assume this.
00:23:37.000 --> 00:23:43.000
And it is only to make the
writing simpler.
00:23:40.000 --> 00:23:46.000
It doesn't really change the
mathematics at all.
00:23:44.000 --> 00:23:50.000
Well, if everything is one,
in order to calculate the
00:23:47.000 --> 00:23:53.000
linearized system,
I'd better use the Jacobian.
00:23:51.000 --> 00:23:57.000
But perhaps it would be better
to write out explicitly what the
00:23:56.000 --> 00:24:02.000
system is.
The system now is x prime
00:23:58.000 --> 00:24:04.000
equals minus x plus xy,
all the coefficients are
00:24:03.000 --> 00:24:09.000
one.
And y prime is equal to y minus
00:24:07.000 --> 00:24:13.000
xy.
What is the Jacobian?
00:24:10.000 --> 00:24:16.000
Well, the Jacobian is the
partial of this with respect to
00:24:14.000 --> 00:24:20.000
x which is minus one plus y,
the partial with respect
00:24:19.000 --> 00:24:25.000
to y which is x,
the partial of this with
00:24:22.000 --> 00:24:28.000
respect to x,
which is minus y,
00:24:24.000 --> 00:24:30.000
and the partial with respect to
y, which is one minus x.
00:24:28.000 --> 00:24:34.000
But I want to evaluate that at
00:24:32.000 --> 00:24:38.000
the point one, one,
00:24:35.000 --> 00:24:41.000
which is the critical point.
It is the critical point
00:24:40.000 --> 00:24:46.000
because I have assumed that all
these parameters have the value
00:24:46.000 --> 00:24:52.000
one.
And when I do that,
00:24:48.000 --> 00:24:54.000
evaluating it at one, one,
00:24:51.000 --> 00:24:57.000
I get what matrix?
Well, I get zero,
00:24:54.000 --> 00:25:00.000
one, negative one,
zero.
00:24:57.000 --> 00:25:03.000
In other words,
00:25:01.000 --> 00:25:07.000
the linearized system is x
prime equals y and y
00:25:07.000 --> 00:25:13.000
prime equals negative x.
00:25:11.000 --> 00:25:17.000
Well, that is just the one we
analyzed before in terms of --
00:25:23.000 --> 00:25:29.000
It's the one whose solutions
are circles.
00:25:25.000 --> 00:25:31.000
In other words,
what we find out is that the
00:25:28.000 --> 00:25:34.000
linearized system is what
geometric type?
00:25:32.000 --> 00:25:38.000
Saddle?
Node?
00:25:34.000 --> 00:25:40.000
Spiral?
None of those.
00:25:37.000 --> 00:25:43.000
It is a center.
The linearized system is a
00:25:44.000 --> 00:25:50.000
center.
00:25:52.000 --> 00:25:58.000
It consists,
in other words,
00:25:54.000 --> 00:26:00.000
of a bunch of curves going
round and round next to each
00:25:58.000 --> 00:26:04.000
other.
Concentric circles,
00:25:59.000 --> 00:26:05.000
in fact.
Well, what is wrong with that?
00:26:03.000 --> 00:26:09.000
Now, we are in deep trouble.
We are now in trouble because
00:26:07.000 --> 00:26:13.000
that is a borderline case.
Let me remind you of what the
00:26:12.000 --> 00:26:18.000
borderline cases are.
When we drew that picture,
00:26:15.000 --> 00:26:21.000
this is from last week's
problem set so you have a
00:26:19.000 --> 00:26:25.000
perfect excuse for forgetting it
totally.
00:26:22.000 --> 00:26:28.000
But I am trying to remind you
of it.
00:26:25.000 --> 00:26:31.000
That is another reason why I am
doing this.
00:26:30.000 --> 00:26:36.000
Remember that picture I asked
you about that appeared on the
00:26:34.000 --> 00:26:40.000
computer screen?
Let's make it a little flatter
00:26:38.000 --> 00:26:44.000
so that I can have room to write
in.
00:26:40.000 --> 00:26:46.000
This is a certain parabola
whose equation you are dying to
00:26:45.000 --> 00:26:51.000
tell me, but I am not asking.
There is the trace,
00:26:48.000 --> 00:26:54.000
this is the determinant,
and the characteristic equation
00:26:53.000 --> 00:26:59.000
is related to the values of
these numbers like plus d equals
00:26:57.000 --> 00:27:03.000
zero.
Then these were spiral synchs.
00:27:02.000 --> 00:27:08.000
This was the region of spiral
sources.
00:27:07.000 --> 00:27:13.000
That is the abbreviation for
sources.
00:27:11.000 --> 00:27:17.000
These were nodal synchs and
these were the nodal sources.
00:27:18.000 --> 00:27:24.000
And down here were the saddles.
And where were the centers?
00:27:24.000 --> 00:27:30.000
The centers were along this
line.
00:27:30.000 --> 00:27:36.000
The centers,
there weren't many of them,
00:27:32.000 --> 00:27:38.000
and they were the separation
for these two regions.
00:27:36.000 --> 00:27:42.000
I will put that down.
These are the centers.
00:27:39.000 --> 00:27:45.000
These other borderlines
correspond to other things.
00:27:42.000 --> 00:27:48.000
These are defective
eigenvalues, zero eigenvalues
00:27:45.000 --> 00:27:51.000
and so on.
But let's concentrate on just
00:27:48.000 --> 00:27:54.000
one of them.
You will find the others
00:27:50.000 --> 00:27:56.000
discussed in the notes,
GS7.
00:27:52.000 --> 00:27:58.000
But if you get this idea then
the rest is just details.
00:27:56.000 --> 00:28:02.000
I think it will be perfectly
clear.
00:28:00.000 --> 00:28:06.000
What is wrong with the center?
The answer is that if we are on
00:28:04.000 --> 00:28:10.000
the center, for example,
this system corresponds to the
00:28:08.000 --> 00:28:14.000
trace being zero and the
determinant being plus one.
00:28:12.000 --> 00:28:18.000
It corresponds to t equals zero
and d equals one.
00:28:16.000 --> 00:28:22.000
This point, in other words.
Now, if I wiggle the
00:28:19.000 --> 00:28:25.000
coefficients of the matrix just
a little bit,
00:28:23.000 --> 00:28:29.000
just change them a little bit,
what I am going to do is move
00:28:27.000 --> 00:28:33.000
off this pink line.
And I might move to here or I
00:28:31.000 --> 00:28:37.000
might move a little way to
there.
00:28:34.000 --> 00:28:40.000
But, if I do that,
I change the geometric type.
00:28:37.000 --> 00:28:43.000
In other words,
being a borderline,
00:28:39.000 --> 00:28:45.000
a slight change of the
parameters can change what it
00:28:42.000 --> 00:28:48.000
changes to.
And, in fact,
00:28:44.000 --> 00:28:50.000
that is geometrically clear.
What does a center look like?
00:28:48.000 --> 00:28:54.000
A center looks like this,
a bunch of curves going around
00:28:51.000 --> 00:28:57.000
all in the same direction,
like concentric circles or
00:28:55.000 --> 00:29:01.000
maybe ellipses or something like
that.
00:28:59.000 --> 00:29:05.000
If I deform the picture just a
little bit, well,
00:29:03.000 --> 00:29:09.000
I might change it into
something that looks like this
00:29:07.000 --> 00:29:13.000
where, after they go around they
don't quite meet up with where
00:29:12.000 --> 00:29:18.000
they were to start with.
And I am going to get a spiral
00:29:17.000 --> 00:29:23.000
synch.
Or, I might do the deformation
00:29:20.000 --> 00:29:26.000
by going around once and I'm a
little outside of where I was.
00:29:25.000 --> 00:29:31.000
In which case it's going to be
a spiral source.
00:29:31.000 --> 00:29:37.000
The fact that just changing
these curves a little bit can
00:29:35.000 --> 00:29:41.000
change the picture to this or to
that corresponds to the fact
00:29:39.000 --> 00:29:45.000
that if you are on here with
this value of t and d,
00:29:43.000 --> 00:29:49.000
zero and one,
and just change t and d a
00:29:46.000 --> 00:29:52.000
little bit, you are going to
move off into these regions.
00:29:51.000 --> 00:29:57.000
You might, of course,
stay on the pink line.
00:29:54.000 --> 00:30:00.000
It is not very likely.
Where does this leave us?
00:29:59.000 --> 00:30:05.000
Well, if the linear system is
not stable in the sense that if
00:30:02.000 --> 00:30:08.000
you change the parameters a
little bit it doesn't change the
00:30:06.000 --> 00:30:12.000
type.
This is, after all,
00:30:07.000 --> 00:30:13.000
only an approximation to the
nonlinear system.
00:30:10.000 --> 00:30:16.000
If in this approximation you
cannot really predict the
00:30:13.000 --> 00:30:19.000
behavior of very well when you
change the parameters,
00:30:17.000 --> 00:30:23.000
then from it you cannot tell
what the original system looked
00:30:20.000 --> 00:30:26.000
like.
In other words,
00:30:23.000 --> 00:30:29.000
the nonlinear system at one,
one might be any one
00:30:28.000 --> 00:30:34.000
of the possibilities,
still a center or it might be a
00:30:32.000 --> 00:30:38.000
spiral synch or it might be a
spiral source.
00:30:36.000 --> 00:30:42.000
Any one of those three is a
possibility.
00:30:39.000 --> 00:30:45.000
It couldn't be a saddle because
that is too far away.
00:30:44.000 --> 00:30:50.000
It couldn't be a node.
That is too far away,
00:30:48.000 --> 00:30:54.000
too.
But it could wander into either
00:30:51.000 --> 00:30:57.000
of these regions and,
therefore, the picture you
00:30:55.000 --> 00:31:01.000
cannot tell which of these three
it is just from this type of
00:31:00.000 --> 00:31:06.000
critical point analysis.
Well, that was Volterra's
00:31:06.000 --> 00:31:12.000
problem.
By the way, the person who
00:31:08.000 --> 00:31:14.000
introduced these equations and
studied them systematically in
00:31:13.000 --> 00:31:19.000
the way in which we are doing it
here was Volterra.
00:31:17.000 --> 00:31:23.000
And, in fact,
he was interested in sharks and
00:31:21.000 --> 00:31:27.000
food fish, as they were called.
What do we do?
00:31:24.000 --> 00:31:30.000
Well, you have to be smart.
What Volterra did was he went
00:31:29.000 --> 00:31:35.000
to method number one.
Let's do it.
00:31:38.000 --> 00:31:44.000
Volterra said I got my
equations x prime equals minus x
00:31:45.000 --> 00:31:51.000
plus xy.
y prime is equal to,
00:31:51.000 --> 00:31:57.000
these are the food fish,
y minus xy.
00:31:58.000 --> 00:32:04.000
Let's eliminate t.
And my problem is,
00:32:03.000 --> 00:32:09.000
of course, I am trying to
determine what type of critical
00:32:07.000 --> 00:32:13.000
point one, one is.
And the method we have used up
00:32:11.000 --> 00:32:17.000
until now has failed because it
gave us a borderline case,
00:32:16.000 --> 00:32:22.000
which is from that we cannot
deduce.
00:32:19.000 --> 00:32:25.000
He said let's eliminate t.
And we get dy over dx
00:32:23.000 --> 00:32:29.000
equals, I am going to factor,
y times one minus x on top.
00:32:29.000 --> 00:32:35.000
And on the bottom factor x
negative one plus y.
00:32:34.000 --> 00:32:40.000
You could solve that before you
came into 18.03,
00:32:39.000 --> 00:32:45.000
right?
This you can separate
00:32:42.000 --> 00:32:48.000
variables.
Let's separate variables.
00:32:46.000 --> 00:32:52.000
The y's all go on one side,
so y goes down here.
00:32:52.000 --> 00:32:58.000
It is (y minus one) over y dy
equals --
00:33:00.000 --> 00:33:06.000
On the other side the dx gets
moved up, one minus x over x dx.
00:33:03.000 --> 00:33:09.000
Now, of course,
you wouldn't dream of using
00:33:06.000 --> 00:33:12.000
partial fractions on this.
It would be illegal because,
00:33:09.000 --> 00:33:15.000
even though it is a rational
function, the quotient of two
00:33:13.000 --> 00:33:19.000
polynomials, the degree of the
top is not smaller than the
00:33:16.000 --> 00:33:22.000
degree of the bottom.
In other words,
00:33:18.000 --> 00:33:24.000
it is a partial fraction so
this degree must be bigger than
00:33:22.000 --> 00:33:28.000
that one.
And it isn't so you would have
00:33:24.000 --> 00:33:30.000
to first divide.
And if you divide by that then
00:33:27.000 --> 00:33:33.000
there is no point in using
partial fractions at all.
00:33:37.000 --> 00:33:43.000
Of course, you would have done
this by basic instant,
00:33:41.000 --> 00:33:47.000
I know.
What is the solution?
00:33:43.000 --> 00:33:49.000
It is y minus log y equals log
x minus x plus some constant.
00:33:49.000 --> 00:33:55.000
That is not the final constant
00:33:53.000 --> 00:33:59.000
I want so I will give it a
subscript one to indicate I want
00:33:58.000 --> 00:34:04.000
more.
This is very hard to see what
00:34:02.000 --> 00:34:08.000
that curve looks like.
We can make it look better by
00:34:05.000 --> 00:34:11.000
exponentiating.
If I exponentiate it,
00:34:08.000 --> 00:34:14.000
going back to high school
mathematics, but I know from
00:34:11.000 --> 00:34:17.000
experience that many of you are
not too good at this step,
00:34:16.000 --> 00:34:22.000
so that is another reason I am
doing it, it will be
00:34:20.000 --> 00:34:26.000
e to the y, times,
that part is easy because
00:34:23.000 --> 00:34:29.000
pluses and minuses change into
times, times e to the negative
00:34:27.000 --> 00:34:33.000
log y.
Well, e to the log y is y.
00:34:31.000 --> 00:34:37.000
If I put a minus sign in front
00:34:35.000 --> 00:34:41.000
of that, that sends it into the
denominator, so instead of being
00:34:40.000 --> 00:34:46.000
y it is one over y
Equals x, e to log x is x,
00:34:44.000 --> 00:34:50.000
e to the negative x
00:34:48.000 --> 00:34:54.000
is, therefore,
times one over e to the x.
00:34:51.000 --> 00:34:57.000
And that is times the
00:34:53.000 --> 00:34:59.000
exponential of c1,
which I will call c2.
00:34:58.000 --> 00:35:04.000
And now if I combine them all
and put them all on one side it
00:35:04.000 --> 00:35:10.000
is x over e to the x times y
over e to the y is equal to yet
00:35:10.000 --> 00:35:16.000
another constant,
one over c2.
00:35:14.000 --> 00:35:20.000
This is my final constant so
00:35:18.000 --> 00:35:24.000
let's call that c.
In other words,
00:35:21.000 --> 00:35:27.000
the integral curves are the
graphs of this equation for
00:35:27.000 --> 00:35:33.000
different values of the constant
c.
00:35:32.000 --> 00:35:38.000
Well, of course you've graphed
an equation like that all the
00:35:37.000 --> 00:35:43.000
time.
What am I going to do with this
00:35:41.000 --> 00:35:47.000
stupid thing?
Well, I deliberately picked
00:35:44.000 --> 00:35:50.000
something which involved all
your learning up until now.
00:35:50.000 --> 00:35:56.000
We are now going into 18.02,
right?
00:35:53.000 --> 00:35:59.000
You looked at that in 18.02 you
would say this is the contour
00:35:59.000 --> 00:36:05.000
curve, these are contour curves
of the function --
00:36:05.000 --> 00:36:11.000
Well, let's write it out the
other way.
00:36:07.000 --> 00:36:13.000
It doesn't make any difference,
but you are more likely to have
00:36:10.000 --> 00:36:16.000
seen the function in this form.
x over e to the x
00:36:13.000 --> 00:36:19.000
you won't recognize,
but this you will.
00:36:15.000 --> 00:36:21.000
In fact, we have had it before
this term.
00:36:18.000 --> 00:36:24.000
It is one of the kinds of
solutions you can add to second
00:36:21.000 --> 00:36:27.000
order equations.
Times y e to the negative y
00:36:23.000 --> 00:36:29.000
equals c.
It is the contour curves of
00:36:26.000 --> 00:36:32.000
this function.
Let's call that function,
00:36:29.000 --> 00:36:35.000
let's say, h of (x,y).
00:36:32.000 --> 00:36:38.000
Well, of course you could throw
it on Matlab,
00:36:35.000 --> 00:36:41.000
as you did in 18.02 maybe,
and ask Matlab to plot the
00:36:39.000 --> 00:36:45.000
function.
But Volterra didn't have that
00:36:42.000 --> 00:36:48.000
luxury.
He was smart,
00:36:43.000 --> 00:36:49.000
too.
So let's be smart instead.
00:36:53.000 --> 00:36:59.000
What is the function x?
Let's use a neutral variable
00:36:56.000 --> 00:37:02.000
like u and plot u e to the
negative u.
00:37:00.000 --> 00:37:06.000
18.01.
At zero, it has the value zero.
00:37:03.000 --> 00:37:09.000
When u is small this has
approximately the value one and,
00:37:08.000 --> 00:37:14.000
therefore, it starts out like
the function u.
00:37:12.000 --> 00:37:18.000
As u goes to infinity,
you know by L'Hopital's rule
00:37:16.000 --> 00:37:22.000
that this goes to zero so it
ends up like this.
00:37:20.000 --> 00:37:26.000
Well, what on earth could it
possibly do except rise to a
00:37:25.000 --> 00:37:31.000
maximum?
But where is that maximum?
00:37:30.000 --> 00:37:36.000
It is easy to see there is a
unique maximum just by 18.01.
00:37:34.000 --> 00:37:40.000
Take the derivative,
find out where the maximum
00:37:37.000 --> 00:37:43.000
point is and you will find,
perhaps you have already done
00:37:41.000 --> 00:37:47.000
it, this is at one.
The maximum occurs at one,
00:37:44.000 --> 00:37:50.000
that is where the derivative is
zero and the value there is one
00:37:49.000 --> 00:37:55.000
times e to the minus one.
00:37:52.000 --> 00:37:58.000
It is one over e,
in other words.
00:37:55.000 --> 00:38:01.000
That is what the curve looks
like.
00:37:57.000 --> 00:38:03.000
What do the contour curves of
this look like?
00:38:02.000 --> 00:38:08.000
Well, if we are looking at the
contour curves of the function
00:38:07.000 --> 00:38:13.000
h, so here is x,
here is y, and we want the
00:38:11.000 --> 00:38:17.000
contour curves of the function h
of (x, y),
00:38:16.000 --> 00:38:22.000
what do I know?
Where is its maximum?
00:38:20.000 --> 00:38:26.000
The maximum point of h of x,
y is where?
00:38:24.000 --> 00:38:30.000
Well, h is the product x e to
the negative x times y e to the
00:38:30.000 --> 00:38:36.000
minus y.
00:38:35.000 --> 00:38:41.000
This has its maximum at one,
this has its maximum at one,
00:38:39.000 --> 00:38:45.000
so where does h of (x,
y) head?
00:38:43.000 --> 00:38:49.000
Well, you have two factors.
One makes each of them the
00:38:48.000 --> 00:38:54.000
biggest they can be.
The maximum must be exactly at
00:38:52.000 --> 00:38:58.000
that critical point one,
one.
00:38:55.000 --> 00:39:01.000
That is our maximum point.
Let's make it conspicuous.
00:39:01.000 --> 00:39:07.000
This is the maximum point.
It is the point one,
00:39:05.000 --> 00:39:11.000
one.
It is the point that makes the
00:39:09.000 --> 00:39:15.000
function biggest.
Now, how about the contour
00:39:13.000 --> 00:39:19.000
curves?
Well, this is the top of the
00:39:16.000 --> 00:39:22.000
mountain.
What is it along the axes?
00:39:19.000 --> 00:39:25.000
Along the axes,
when either x or y is zero,
00:39:23.000 --> 00:39:29.000
the function has the value
zero, so it is biggest there.
00:39:28.000 --> 00:39:34.000
It has the value zero here.
So it is zero value.
00:39:33.000 --> 00:39:39.000
What are the contour curves?
Well, we must have a mountain
00:39:38.000 --> 00:39:44.000
peak here.
This is the unique maximum
00:39:41.000 --> 00:39:47.000
point.
In fact, it is easy to see the
00:39:44.000 --> 00:39:50.000
contour curves just surround it
like that.
00:39:47.000 --> 00:39:53.000
Now, the reason they cannot be
spirals, well,
00:39:51.000 --> 00:39:57.000
in the first place,
you never see contour curves
00:39:55.000 --> 00:40:01.000
that were spirals.
That is a good reason.
00:40:00.000 --> 00:40:06.000
It is a mountain.
That is the way contour curves
00:40:03.000 --> 00:40:09.000
of a mountain look.
But, all right,
00:40:06.000 --> 00:40:12.000
that is not a good argument.
But notice that along each
00:40:10.000 --> 00:40:16.000
horizontal line here,
I want to know how many times
00:40:14.000 --> 00:40:20.000
can it intersect the contour
curve?
00:40:16.000 --> 00:40:22.000
That is the same as asking
along one of these lines how
00:40:20.000 --> 00:40:26.000
many times could it intersect?
See, this is a graph of the
00:40:25.000 --> 00:40:31.000
function along this horizontal
line.
00:40:29.000 --> 00:40:35.000
And, therefore,
how many times can it intersect
00:40:32.000 --> 00:40:38.000
one of the contour curves the
same number of times that a
00:40:36.000 --> 00:40:42.000
horizontal line can intersect
this curve?
00:40:39.000 --> 00:40:45.000
Twice.
Only once if you are exactly at
00:40:42.000 --> 00:40:48.000
that height and after that
never.
00:40:44.000 --> 00:40:50.000
But here it can intersect each
contour curve only twice which
00:40:48.000 --> 00:40:54.000
means these things cannot be
spirals.
00:40:51.000 --> 00:40:57.000
Otherwise, they would intersect
those horizontal and vertical
00:40:55.000 --> 00:41:01.000
lines many times instead of just
twice.
00:41:00.000 --> 00:41:06.000
That is what it looks like.
In fact, we can even put in the
00:41:04.000 --> 00:41:10.000
direction without any effort.
We know that the sharks die out
00:41:09.000 --> 00:41:15.000
without any food fish and the
food fish grow.
00:41:13.000 --> 00:41:19.000
Well, I think this has to be
clockwise.
00:41:16.000 --> 00:41:22.000
Of course, the guys nearby must
be going the same way.
00:41:21.000 --> 00:41:27.000
And, therefore,
the guys near them must be
00:41:24.000 --> 00:41:30.000
going the same way.
And it is sort of a domino
00:41:28.000 --> 00:41:34.000
effect.
The direction spreads and there
00:41:32.000 --> 00:41:38.000
is only one compatible way of
making these.
00:41:35.000 --> 00:41:41.000
They all must be going
clockwise.
00:41:37.000 --> 00:41:43.000
What is happening,
actually?
00:41:39.000 --> 00:41:45.000
At this point there are a lot
of sharks but not so many food
00:41:44.000 --> 00:41:50.000
fish.
The sharks eat the few
00:41:46.000 --> 00:41:52.000
remaining food fish,
and now the sharks start to die
00:41:49.000 --> 00:41:55.000
out because they have nothing to
eat.
00:41:52.000 --> 00:41:58.000
When they are practically gone,
the food fish can start growing
00:41:56.000 --> 00:42:02.000
again and grow and grow.
For a while they grow happily,
00:42:02.000 --> 00:42:08.000
and then the sharks suddenly
start being able to find them
00:42:06.000 --> 00:42:12.000
again.
And the few remaining sharks
00:42:09.000 --> 00:42:15.000
start eating them,
the population of sharks grows,
00:42:13.000 --> 00:42:19.000
the food fish start dying out
again, and the cycle starts all
00:42:18.000 --> 00:42:24.000
over again.
Now, I wanted to talk about the
00:42:21.000 --> 00:42:27.000
qualitative behavior.
This is, in some ways,
00:42:24.000 --> 00:42:30.000
the most important part of the
lecture.
00:42:29.000 --> 00:42:35.000
I wanted to discuss,
just as we did at the beginning
00:42:35.000 --> 00:42:41.000
of the term, the effect of
fishing with nets at a constant
00:42:43.000 --> 00:42:49.000
rate k.
00:42:51.000 --> 00:42:57.000
Just as near the beginning of
the term, we talked about
00:42:54.000 --> 00:43:00.000
fishing a single population.
The only difference is,
00:42:57.000 --> 00:43:03.000
now there are two populations.
Both sharks and the food fish.
00:43:02.000 --> 00:43:08.000
Now, what happens to the
equations?
00:43:05.000 --> 00:43:11.000
Well, the equations start out
just as they were.
00:43:10.000 --> 00:43:16.000
I am now reverting to a and b.
You will see why.
00:43:14.000 --> 00:43:20.000
It's because we need the
general coefficients.
00:43:18.000 --> 00:43:24.000
Plus bxy.
But, if we are fishing with
00:43:21.000 --> 00:43:27.000
nets, then a certain fraction of
the sharks in the sea get hauled
00:43:27.000 --> 00:43:33.000
in by the net.
And say the fishing rate is k.
00:43:32.000 --> 00:43:38.000
There is a fishing term.
We remove minus k times a
00:43:36.000 --> 00:43:42.000
certain fraction of the sharks
in the sea.
00:43:40.000 --> 00:43:46.000
How about the food fish?
Well, we remove them,
00:43:44.000 --> 00:43:50.000
too.
And we don't distinguish
00:43:47.000 --> 00:43:53.000
between sharks and food fish.
This is cy minus dxy.
00:43:51.000 --> 00:43:57.000
But we also remove a certain
00:43:54.000 --> 00:44:00.000
fraction of them.
Minus k times y.
00:43:58.000 --> 00:44:04.000
What, therefore,
is the new system?
00:44:03.000 --> 00:44:09.000
The new system is therefore x
prime equals minus a plus k.
00:44:10.000 --> 00:44:16.000
I am combining those two x
terms.
00:44:14.000 --> 00:44:20.000
That is times x.
Plus bxy.
00:44:17.000 --> 00:44:23.000
And
the y prime is c minus k times y
00:44:24.000 --> 00:44:30.000
minus dxy.
00:44:30.000 --> 00:44:36.000
To solve these,
I do not have to go through the
00:44:33.000 --> 00:44:39.000
analysis.
All I have to do is change the
00:44:37.000 --> 00:44:43.000
numbers, change these
coefficients from a to a plus k
00:44:41.000 --> 00:44:47.000
and c minus k.
The old critical point was the
00:44:46.000 --> 00:44:52.000
point c over d,
a over b.
00:44:49.000 --> 00:44:55.000
The new critical point with
00:44:52.000 --> 00:44:58.000
fishing is what?
Well, the parameters have been
00:44:56.000 --> 00:45:02.000
changed by the addition of k.
The new critical point is c
00:45:02.000 --> 00:45:08.000
minus k over d and a plus k over
d.
00:45:07.000 --> 00:45:13.000
What is the effect of fishing?
00:45:11.000 --> 00:45:17.000
Well, if the old critical point
was over here,
00:45:15.000 --> 00:45:21.000
let's say this is the point c
over d and this is the
00:45:21.000 --> 00:45:27.000
point a over b,
that is the old critical point
00:45:27.000 --> 00:45:33.000
that was over here.
The result of fishing is to
00:45:33.000 --> 00:45:39.000
lower the value of x and raise
the value here.
00:45:38.000 --> 00:45:44.000
The new critical point is,
this gets lowered to c minus k
00:45:44.000 --> 00:45:50.000
over d,
and this gets raised to a plus
00:45:50.000 --> 00:45:56.000
k over b.
In other words,
00:45:54.000 --> 00:46:00.000
the new point is there.
The effect of fishing is to --
00:46:02.000 --> 00:46:08.000
It does not treat the sharks
and the yumfish equally.
00:46:07.000 --> 00:46:13.000
The effect of fishing lowers
the shark population.
00:46:11.000 --> 00:46:17.000
See, the critical point gives
sort of the average shark
00:46:17.000 --> 00:46:23.000
population.
Of course, it cycles around
00:46:20.000 --> 00:46:26.000
these.
But, on the average,
00:46:23.000 --> 00:46:29.000
this gives how many sharks
there are and how many food fish
00:46:28.000 --> 00:46:34.000
there are.
The new critical point with
00:46:33.000 --> 00:46:39.000
fishing lowers the shark
population and raises the food
00:46:39.000 --> 00:46:45.000
fish population.
That is not intuitive.
00:46:42.000 --> 00:46:48.000
And, in fact,
that was observed
00:46:45.000 --> 00:46:51.000
experimentally at a slightly
different context.
00:46:50.000 --> 00:46:56.000
And that is why Volterra
started working on the problem.
00:46:55.000 --> 00:47:01.000
I will need three more minutes.
The most interesting
00:47:01.000 --> 00:47:07.000
application of all is not to
sharks and food fish.
00:47:03.000 --> 00:47:09.000
I cannot assume that you are
dramatically interested in how
00:47:07.000 --> 00:47:13.000
many of them there are in the
ocean, but you might be more
00:47:10.000 --> 00:47:16.000
interested in this.
00:47:20.000 --> 00:47:26.000
That thing about lowering is
called Volterra's principle.
00:47:25.000 --> 00:47:31.000
Put that in your books.
Volterra's principle.
00:47:29.000 --> 00:47:35.000
Volterra is spelt over there.
00:47:43.000 --> 00:47:49.000
-- has found more modern
applications than sharks.
00:47:48.000 --> 00:47:54.000
Suppose you consider fish in a
pond.
00:47:53.000 --> 00:47:59.000
You have mosquito larvae which
breed in the pond.
00:47:58.000 --> 00:48:04.000
This happens.
And then suddenly there is a
00:48:03.000 --> 00:48:09.000
plague of mosquitoes and
concerned citizens.
00:48:07.000 --> 00:48:13.000
And this is what happened in
the '50s.
00:48:11.000 --> 00:48:17.000
That was before you were born,
but not before I was.
00:48:16.000 --> 00:48:22.000
What happened was a lot of
mosquitoes.
00:48:19.000 --> 00:48:25.000
Everybody said the mosquitoes
breed in the little stagnant
00:48:25.000 --> 00:48:31.000
ponds so spray DDT on them.
DDT them.
00:48:30.000 --> 00:48:36.000
Dump it in the ponds.
That will kill all the larvae
00:48:34.000 --> 00:48:40.000
and we won't get bitten anymore.
When you put DDT in the pond,
00:48:39.000 --> 00:48:45.000
as people did not realize at
the time because these things
00:48:44.000 --> 00:48:50.000
were new, of course you kill the
mosquitoes, but you also kill
00:48:49.000 --> 00:48:55.000
the fish because DDT is
poisonous to fish.
00:48:53.000 --> 00:48:59.000
What, in effect,
you are doing mathematically is
00:48:57.000 --> 00:49:03.000
the same as harvesting the fish
population with the sharks and
00:49:02.000 --> 00:49:08.000
the food fish.
The result is,
00:49:06.000 --> 00:49:12.000
the fish are the predators
because they eat the mosquito
00:49:11.000 --> 00:49:17.000
larvae, big food,
there was a certain
00:49:15.000 --> 00:49:21.000
equilibrium, according to
Volterra's principle.
00:49:20.000 --> 00:49:26.000
With DDT that equilibrium moves
to here.
00:49:24.000 --> 00:49:30.000
In other words,
the effect of indiscriminately
00:49:28.000 --> 00:49:34.000
spraying the pond with DDT is to
increase the number of
00:49:34.000 --> 00:49:40.000
mosquitoes and kill fish.
And, in fact,
00:49:39.000 --> 00:49:45.000
that is exactly what was
observed.
00:49:42.000 --> 00:49:48.000
The same thing was observed
with the bird population and
00:49:48.000 --> 00:49:54.000
insects.
Spraying trees for insects to
00:49:51.000 --> 00:49:57.000
get rid of some pests ends up
killing more birds than it does
00:49:58.000 --> 00:50:04.000
insects and the insects
increase.
00:50:01.000 --> 00:50:07.000
Thanks.