WEBVTT

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As a matter of fact,
it plots them very accurately.

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But it is something you also
need to learn to do yourself,

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as you will see when we study
nonlinear equations.

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It is a skill.
And since a couple of important

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mathematical ideas are involved
in it, I think it is a very good

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thing to spend just a little
time on, one lecture in fact,

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plus a little more on the
problem set that I will give

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out.
The last problem set that I

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will give out on Friday.
I thought it might be a little

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more fun to, again,
have a simple-minded model.

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No romance this time.
We are going to have a little

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model of war,
but I have made it sort of

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sublimated war.
Let's take as the system,

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I am going to let two of those
be parameters,

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you know, be variable,
in other words.

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And the other two I will keep
fixed, so that you can

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concentrate on them better.
I will take a and d to be

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negative 1 and negative 3.
And the other ones we will

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leave open, so let's call this
one b times y,

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and this other one will be c
times x.

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I am going to model this as a
fight between two states,

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both of which are trying to
attract tourists.

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Let's say this is Massachusetts
and this will be New Hampshire,

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its enemy to the North.
Both are busy advertising these

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days on television.
People are making their summer

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plans.
Come to New Hampshire,

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you know, New Hampshire has
mountains and Massachusetts has

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quaint little fishing villages
and stuff like that.

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So what are these numbers?
Well, first of all,

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what do x and y represent?
x and y basically are the

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advertising budgets for tourism,
you know, the amount each state

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plans to spend during the year.
However, I do not want zero

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value to mean they are not
spending anything.

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It represents departure from
the normal equilibrium.

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x and y represent departures --

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-- from the normal amount of
money they spend advertising for

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tourists.
The normal tourist advertising

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budget.

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If they are both zero,
it means that both states are

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spending what they normally
spend in that year.

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If x is positive,
it means that Massachusetts has

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decided to spend more in the
hope of attracting more tourists

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and if negative spending less.
What is the significance of

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these two coefficients?
Those are the normal things

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which return you to equilibrium.
In other words,

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if x gets bigger than normal,
if Massachusetts spends more

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there is a certain poll to spend
less because we are wasting all

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this money on the tourists that
are not going to come when we

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could be spending it on
education or something like

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that.
If x gets to be negative,

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the governor tries to spend
less.

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Then all the local city Chamber
of Commerce rise up and start

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screaming that our economy is
going to go bankrupt because we

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won't get enough tourists and
that is because you are not

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spending enough money.
There is a push to always

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return it to the normal,
and that is what this negative

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sign means.
The same thing for New

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Hampshire.
What does it mean that this is

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negative three and that is
negative one?

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It just means that the Chamber
of Commerce yells three times as

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loudly in New Hampshire.
It is more sensitive,

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in other words,
to changes in the budget.

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Now, how about the other?
Well, these represent the

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war-like features of the
situation.

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Normally these will be positive
numbers.

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Because when Massachusetts sees
that New Hampshire has budgeted

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this year more than its normal
amount, the natural instinct is

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we are fighting.
This is war.

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This is a positive number.
We have to budget more,

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too.
And the same thing for New

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Hampshire.
The size of these coefficients

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gives you the magnitude of the
reaction.

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If they are small Massachusetts
say, well, they are spending

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more but we don't have to follow
them.

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We will bucket a little bit.
If it is a big number then oh,

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my God, heads will roll.
We have to triple them and put

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them out of business.
This is a model,

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in fact, for all sorts of
competition.

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It was used for many years to
model in simper times armaments

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races between countries.
It is certainly a simple-minded

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model for any two companies in
competition with each other if

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certain conditions are met.
Well, what I would like to do

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now is try different values of
those numbers.

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And, in each case,
show you how to sketch the

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solutions at different cases.
And then, for each different

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case, we will try to interpret
if it makes sense or not.

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My first set of numbers is,
the first case is --

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-- x prime equals negative x
plus 2y.

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And y prime equals,
this is going to be zero,

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so it is simply minus
3 times y.

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Now, what does this mean?
Well, this means that

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Massachusetts is behaving
normally, but New Hampshire is a

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very placid state,
and the governor is busy doing

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other things.
And people say Massachusetts is

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spending more this year,
and the Governor says,

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so what.
The zero is the so what factor.

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In other words,
we are not going to respond to

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them.
We will do our own thing.

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What is the result of this?
Is Massachusetts going to win

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out?
What is going to be the

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ultimate effect on the budget?
Well, what we have to do is,

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so the program is first let's
quickly solve the equations

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using a standard technique.
I am just going to make marks

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on the board and trust to the
fact that you have done enough

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of this yourself by now that you
know what the marks mean.

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I am not going to label what
everything is.

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I am just going to trust to
luck.

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The matrix A is negative 1,
2, 0, negative 3.

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The characteristic equation,

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the second coefficient is the
trace, which is minus 4,

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but you have to change its
sign, so that makes it plus 4.

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And the constant term is the
determinant, which is 3 minus 0,

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so that is plus 3 equals zero.
This factors into lambda plus 3

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times lambda plus one.
And it means the roots

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therefore are,
one root is lambda equals

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negative 3 and the other root is
lambda equals negative 1.

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These are the eigenvalues.
With each eigenvalue goes an

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eigenvector.
The eigenvector is found by

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solving an equation for the
coefficients of the eigenvector,

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the components of the
eigenvector.

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Here I used negative 1 minus
negative 3, which makes 2.

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The first equation is 2a1 plus
2a2 is equal to zero.

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The second one will be,
in fact, in this case simply

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0a1 plus 0a2 so it won't give me
any information at all.

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That is not what usually
happens, but it is what happens

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in this case.
What is the solution?

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The solution is the vector
alpha equals,

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well, 1, negative 1
would be a good thing to

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use.
That is the eigenvector,

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so this is the e-vector.
How about lambda equals

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negative 1?
Let's give it a little more

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room.
If lambda is negative 1 then

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here I put negative 1 minus
negative 1.

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That makes zero.
I will write in the zero

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because this is confusing.
It is zero times a1.

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And the next coefficient is 2
a2, is zero.

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People sometimes go bananas
over this, in spite of the fact

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that this is the easiest
possible case you can get.

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I guess if they go bananas over
it, it proves it is not all that

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easy, but it is easy.
What now is the eigenvector

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that goes with this?
Well, this term isn't there.

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It is zero.
The equation says that a2 has

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to be zero.
And it doesn't say anything

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about a1, so let's make it 1.

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Now, out of this data,
the final step is to make the

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general solution.
What is it?

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(x, y) equals,
well, a constant times the

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first normal mode.
The solution constructed from

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the eigenvalue and the
eigenvector.

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That is going to be 1,
negative 1 e to the minus 3t.

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And then the other normal mode

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times an arbitrary constant will
be (1, 0) times e to the

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negative t.

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The lambda is this factor which
produces that,

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of course.
Now, one way of looking at it

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is, first of all,
get clearly in your head this

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is a pair of parametric
equations just like what you

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studied in 18.02.
Let's write them out explicitly

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just this once.
x equals c1 times e to the

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negative 3t plus c2 times e to
the negative t.

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And what is y?

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y is equal to minus c1 e to the
minus 3t plus zero.

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I can stop there.

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In some sense,
all I am asking you to do is

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plot that curve.
In the x,y-plane,

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plot the curve given by this
pair of parametric equations.

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And you can choose your own
values of c1,

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c2.
For different values of c1 and

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c2 there will be different
curves.

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Give me a feeling for what they
all look like.

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Well, I think most of you will
recognize you didn't have stuff

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like this.
These weren't the kind of

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curves you plotted.
When you did parametric

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equations in 18.02,
you did stuff like x equals

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cosine t, y equals sine t.

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Everybody knows how to do that.
A few other curves which made

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lines or nice things,
but nothing that ever looked

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like that.
And so the computer will plot

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it by actually calculating
values but, of course,

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we will not.
That is the significance of the

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word sketch.
I am not asking you to plot

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carefully, but to give me some
general geometric picture of

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what all these curves look like
without doing any work.

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Without doing any work.
Well, that sounds promising.

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Okay, let's try to do it
without doing any work.

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Where shall I begin?
Hidden in this formula are four

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solutions that are extremely
easy to plot.

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So begin with the four easy
solutions, and then fill in the

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rest.
Now, which are the easy

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solutions?
The easy solutions are c1

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equals plus or minus 1,
c2 equals zero,

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or c1 equals zero,
or c1 = 0, c2 equals plus or

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minus 1.
By choosing those four values

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of c1 and c2,
I get simple solutions

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corresponding to the normal
mode.

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If c1 is one and c2 is zero,
I am talking about (1,

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negative 1) e to the minus 3t,

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and that is very easy plot.
Let's start plotting them.

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What I am going to do is
color-code them so you will be

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able to recognize what it is I
am plotting.

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Let's see.
What colors should we use?

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We will use pink and orange.
This will be our pink solution

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and our orange solution will be
this one.

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Let's plot the pink solution
first.

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The pink solution corresponds
to c1 equals 1 and c2

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equals zero.
Now, that solution looks like--

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Let's write it in pink.

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No, let's not write it in pink.
What is the solution?

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It looks like x equals e to the
negative 3t,

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y equals minus e to the minus
3t.

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Well, that's not a good way to
look at it, actually.

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The best way to look at it is
to say at t equals zero,

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where is it?
It is at the point 1,

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negative 1.

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And what is it doing as t
increases?

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Well, it keeps the direction,
but travels.

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The amplitude,
the distance from the origin

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keeps shrinking.
As t increases,

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this factor,
so it is the tip of this

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vector, except the vector is
shrinking.

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It is still in the direction of
1, negative 1,

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but it is shrinking in
length because its amplitude is

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shrinking according to the law e
to the negative 3t.

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In other words,
this curve looks like this.

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At t equals zero it is over
here, and it goes along this

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diagonal line until as t equals
infinity, it gets to infinity,

00:16:14.000 --> 00:16:20.000
it reaches the origin.
Of course, it never gets there.

00:16:18.000 --> 00:16:24.000
It goes slower and slower and
slower in order that it may

00:16:23.000 --> 00:16:29.000
never reach the origin.
What was it doing for values of

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t less than zero?
The same thing,

00:16:31.000 --> 00:16:37.000
except it was further away.
It comes in from infinity along

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that straight line.
In other words,

00:16:37.000 --> 00:16:43.000
the eigenvector determines the
line on which it travels and the

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eigenvalue determines which way
it goes.

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If the eigenvalue is negative,
it is approaching the origin as

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t increases.
How about the other one?

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Well, if c1 is negative 1,
then everything is the

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same except it is the mirror
image of this one.

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If c1 is negative 1,
then at t equals zero it is at

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this point.
And, once again,

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the same reasoning shows that
it is coming into the origin as

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t increases.
I have now two solutions,

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this one corresponding to c1
equals 1,

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and the other one c2 equals
zero.

00:17:19.000 --> 00:17:25.000
This one corresponds to c1
equals negative 1.

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How about the other guy,
the orange guy?

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Well, now c1 is zero,
c2 is one, let's say.

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It is the vector (1,
0), but otherwise everything is

00:17:33.000 --> 00:17:39.000
the same.
I start now at the point (1,

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0) at time zero.
And, as t increases,

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I come into the origin always
along that direction.

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And before that I came in from
infinity.

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And, again, if c2 is 1
and if c2 is negative 1,

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I do the same thing but
on the other side.

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That wasn't very hard.
I plotted four solutions.

00:18:04.000 --> 00:18:10.000
And now I roll up my sleeves
and waive my hands to try to get

00:18:10.000 --> 00:18:16.000
others.
The general philosophy is the

00:18:14.000 --> 00:18:20.000
following.
The general philosophy is the

00:18:18.000 --> 00:18:24.000
differential equation looks like
this.

00:18:21.000 --> 00:18:27.000
It is a system of differential
equations.

00:18:25.000 --> 00:18:31.000
These are continuous functions.
That means when I draw the

00:18:31.000 --> 00:18:37.000
velocity field corresponding to
that system of differential

00:18:36.000 --> 00:18:42.000
equations, because their
functions are continuous,

00:18:39.000 --> 00:18:45.000
as I move from one (x,
y) point to another the

00:18:43.000 --> 00:18:49.000
direction of the velocity
vectors change continuously.

00:18:46.000 --> 00:18:52.000
It never suddenly reverses
without something like that.

00:18:50.000 --> 00:18:56.000
Now, if that changes
continuously then the

00:18:53.000 --> 00:18:59.000
trajectories must change
continuously,

00:18:56.000 --> 00:19:02.000
too.
In other words,

00:18:59.000 --> 00:19:05.000
nearby trajectories should be
doing approximately the same

00:19:03.000 --> 00:19:09.000
thing.
Well, that means all the other

00:19:05.000 --> 00:19:11.000
trajectories are ones which come
like that must be going also

00:19:10.000 --> 00:19:16.000
toward the origin.
If I start here,

00:19:12.000 --> 00:19:18.000
probably I have to follow this
one.

00:19:15.000 --> 00:19:21.000
They are all coming to the
origin, but that is a little too

00:19:19.000 --> 00:19:25.000
vague.
How do they come to the origin?

00:19:22.000 --> 00:19:28.000
In other words,
are they coming in straight

00:19:25.000 --> 00:19:31.000
like that?
Probably not.

00:19:26.000 --> 00:19:32.000
Then what are they doing?
Now we are coming to the only

00:19:32.000 --> 00:19:38.000
point in the lecture which you
might find a little difficult.

00:19:36.000 --> 00:19:42.000
Try to follow what I am doing
now.

00:19:38.000 --> 00:19:44.000
If you don't follow,
it is not well done in the

00:19:42.000 --> 00:19:48.000
textbook, but it is very well
done in the notes because I

00:19:46.000 --> 00:19:52.000
wrote them myself.
Please, it is done very

00:19:49.000 --> 00:19:55.000
carefully in the notes,
patiently follow through the

00:19:52.000 --> 00:19:58.000
explanation.
It takes about that much space.

00:19:55.000 --> 00:20:01.000
It is one of the important
ideas that your engineering

00:19:59.000 --> 00:20:05.000
professors will expect you to
understand.

00:20:04.000 --> 00:20:10.000
Anyway, I know this only from
the negative one because they

00:20:08.000 --> 00:20:14.000
say to me at lunch,
ruin my lunch by saying I said

00:20:12.000 --> 00:20:18.000
it to my students and got
nothing but blank looks.

00:20:16.000 --> 00:20:22.000
What do you guys teach them
over there?

00:20:19.000 --> 00:20:25.000
Blah, blah, blah.
Maybe we ought to start

00:20:22.000 --> 00:20:28.000
teaching it ourselves.
Sure.

00:20:25.000 --> 00:20:31.000
Why don't they start cutting
their own hair,

00:20:28.000 --> 00:20:34.000
too?

00:20:35.000 --> 00:20:41.000
Here is the idea.
Let me recopy that solution.

00:20:40.000 --> 00:20:46.000
The solution looks like (1,
negative 1) e to the minus 3t

00:20:46.000 --> 00:20:52.000
plus c2, (1, 0) e to the
negative t.

00:20:56.000 --> 00:21:02.000
What I ask is as t goes to
infinity, I feel sure that the

00:21:00.000 --> 00:21:06.000
trajectories must be coming into
the origin because these guys

00:21:04.000 --> 00:21:10.000
are doing that.
And, in fact,

00:21:06.000 --> 00:21:12.000
that is confirmed.
As t goes to infinity,

00:21:09.000 --> 00:21:15.000
this goes to zero and that goes
to zero regardless of what the

00:21:13.000 --> 00:21:19.000
c1 and c2 are.
That makes it clear that this

00:21:17.000 --> 00:21:23.000
goes to zero no matter what the
c1 and c2 are as t goes to

00:21:21.000 --> 00:21:27.000
infinity, but I would like to
analyze it a little more

00:21:25.000 --> 00:21:31.000
carefully.
As t goes to infinity,

00:21:28.000 --> 00:21:34.000
I have the sum of two terms.
And what I ask is,

00:21:32.000 --> 00:21:38.000
which term is dominant?
Of these two terms,

00:21:36.000 --> 00:21:42.000
are they of equal importance,
or is one more important than

00:21:41.000 --> 00:21:47.000
the other?
When t is 10,

00:21:43.000 --> 00:21:49.000
for example,
that is not very far on the way

00:21:47.000 --> 00:21:53.000
to infinity, but it is certainly
far enough to illustrate.

00:21:52.000 --> 00:21:58.000
Well, e to the minus 10
is an extremely

00:21:56.000 --> 00:22:02.000
small number.
The only thing smaller is e to

00:22:01.000 --> 00:22:07.000
the minus 30.
The term that dominates,

00:22:05.000 --> 00:22:11.000
they are both small,
but relatively-speaking this

00:22:08.000 --> 00:22:14.000
one is much larger because this
one only has the factor e to the

00:22:13.000 --> 00:22:19.000
minus 10,
whereas, this has the factor e

00:22:17.000 --> 00:22:23.000
to the minus 30,
which is vanishingly small.

00:22:22.000 --> 00:22:28.000
In other words,
as t goes to infinity --

00:22:26.000 --> 00:22:32.000
Well, let's write it the other
way.

00:22:28.000 --> 00:22:34.000
This is the dominant term,
as t goes to infinity.

00:22:38.000 --> 00:22:44.000
Now, just the opposite is true
as t goes to minus infinity.

00:22:43.000 --> 00:22:49.000
t going to minus infinity means
I am backing up along these

00:22:48.000 --> 00:22:54.000
curves.
As t goes to minus infinity,

00:22:51.000 --> 00:22:57.000
let's say t gets to be negative
100, this is e to the 100,

00:22:56.000 --> 00:23:02.000
but this is e to the 300,

00:23:01.000 --> 00:23:07.000
which is much,
much bigger.

00:23:03.000 --> 00:23:09.000
So this is the dominant term as
t goes to negative infinity.

00:23:18.000 --> 00:23:24.000
Now what I have is the sum of
two vectors.

00:23:20.000 --> 00:23:26.000
Let's first look at what
happens as t goes to infinity.

00:23:24.000 --> 00:23:30.000
As t goes to infinity,
I have the sum of two vectors.

00:23:28.000 --> 00:23:34.000
This one is completely
negligible compared with the one

00:23:31.000 --> 00:23:37.000
on the right-hand side.
In other words,

00:23:35.000 --> 00:23:41.000
for a all intents and purposes,
as t goes to infinity,

00:23:38.000 --> 00:23:44.000
it is this thing that takes
over.

00:23:41.000 --> 00:23:47.000
Therefore, what does the
solution look like as t goes to

00:23:45.000 --> 00:23:51.000
infinity?
The answer is it follows the

00:23:47.000 --> 00:23:53.000
yellow line.
Now, what does it look like as

00:23:50.000 --> 00:23:56.000
it backs up?
As it came in from negative

00:23:53.000 --> 00:23:59.000
infinity, what does it look
like?

00:23:56.000 --> 00:24:02.000
Now, this one is a little
harder to see.

00:24:00.000 --> 00:24:06.000
This is big,
but this is infinity bigger.

00:24:03.000 --> 00:24:09.000
I mean very,
very much bigger,

00:24:06.000 --> 00:24:12.000
when t is a large negative
number.

00:24:09.000 --> 00:24:15.000
Therefore, what I have is the
sum of a very big vector.

00:24:14.000 --> 00:24:20.000
You're standing on the moon
looking at the blackboard,

00:24:19.000 --> 00:24:25.000
so this is really big.
This is a very big vector.

00:24:24.000 --> 00:24:30.000
This is one million meters
long, and this is only 20

00:24:29.000 --> 00:24:35.000
meters long.
That is this guy,

00:24:33.000 --> 00:24:39.000
and that is this guy.
I want the sum of those two.

00:24:36.000 --> 00:24:42.000
What does the sum look like?
The answer is a sum is

00:24:40.000 --> 00:24:46.000
approximately parallel to the
long guy because this is

00:24:44.000 --> 00:24:50.000
negligible.
This does not mean they are

00:24:47.000 --> 00:24:53.000
next to each other.
They are slightly tilted over,

00:24:51.000 --> 00:24:57.000
but not very much.
In other words,

00:24:53.000 --> 00:24:59.000
as t goes to negative infinity
it doesn't coincide with this

00:24:58.000 --> 00:25:04.000
vector.
The solution doesn't,

00:25:01.000 --> 00:25:07.000
but it is parallel to it.
It has the same direction.

00:25:05.000 --> 00:25:11.000
I am done.
It means far away from the

00:25:07.000 --> 00:25:13.000
origin, it should be parallel to
the pink line.

00:25:11.000 --> 00:25:17.000
Near the origin it should turn
and become more or less

00:25:15.000 --> 00:25:21.000
coincident with the orange line.
And those were the solutions.

00:25:19.000 --> 00:25:25.000
That's how they look.

00:25:27.000 --> 00:25:33.000
How about down here?
The same thing,

00:25:30.000 --> 00:25:36.000
like that, but then after a
while they turn and join.

00:25:35.000 --> 00:25:41.000
Here, they have to turn around
to join up, but they join.

00:25:41.000 --> 00:25:47.000
And that is,
in a simple way,

00:25:44.000 --> 00:25:50.000
the sketches of those
functions.

00:25:47.000 --> 00:25:53.000
That is how they must look.
What does this say about our

00:25:53.000 --> 00:25:59.000
state?
Well, it says that the fact

00:25:57.000 --> 00:26:03.000
that the governor of New
Hampshire is indifferent to what

00:26:01.000 --> 00:26:07.000
Massachusetts is doing produces
ultimately harmony.

00:26:06.000 --> 00:26:12.000
Both states revert ultimately
their normal advertising budgets

00:26:11.000 --> 00:26:17.000
in spite of the fact that
Massachusetts is keeping an eye

00:26:15.000 --> 00:26:21.000
peeled out for the slightest
misbehavior on the part of New

00:26:20.000 --> 00:26:26.000
Hampshire.
Peace reins,

00:26:22.000 --> 00:26:28.000
in other words.
Now you should know some names.

00:26:27.000 --> 00:26:33.000
Let's see.
I will write names in purple.

00:26:30.000 --> 00:26:36.000
There are two words that are
used to describe this situation.

00:26:35.000 --> 00:26:41.000
First is the word that
describes the general pattern of

00:26:40.000 --> 00:26:46.000
the way these lines look.
The word for that is a node.

00:26:44.000 --> 00:26:50.000
And the fact that all the
trajectories end up at the

00:26:48.000 --> 00:26:54.000
origin for that one uses the
word sink.

00:26:52.000 --> 00:26:58.000
This could be modified to nodal
sink.

00:26:55.000 --> 00:27:01.000
That would be better.
Nodal sink, let's say.

00:27:00.000 --> 00:27:06.000
Nodal sink or,
if you like to write them in

00:27:03.000 --> 00:27:09.000
the opposite order,
sink node.

00:27:06.000 --> 00:27:12.000
In the same way there would be
something called a source node

00:27:11.000 --> 00:27:17.000
if I reversed all the arrows.
I am not going to calculate an

00:27:16.000 --> 00:27:22.000
example.
Why don't I simply do it by

00:27:19.000 --> 00:27:25.000
giving you --
For example,

00:27:23.000 --> 00:27:29.000
if the matrix A produced a
solution instead of that one.

00:27:28.000 --> 00:27:34.000
Suppose it looked like 1,
negative 1 e to the 3t.

00:27:32.000 --> 00:27:38.000
The eigenvalues were reversed,

00:27:36.000 --> 00:27:42.000
were now positive.
And I will make the other one

00:27:41.000 --> 00:27:47.000
positive, too.
c2 1, 0 e to the t.

00:27:57.000 --> 00:28:03.000
What would that change in the
picture?

00:27:59.000 --> 00:28:05.000
The answer is essentially
nothing, except the direction of

00:28:04.000 --> 00:28:10.000
the arrows.
In other words,

00:28:06.000 --> 00:28:12.000
the first thing would still be
1, negative 1.

00:28:09.000 --> 00:28:15.000
The only difference is that now

00:28:12.000 --> 00:28:18.000
as t increases we go the other
way.

00:28:15.000 --> 00:28:21.000
And here the same thing,
we have still the same basic

00:28:19.000 --> 00:28:25.000
vector, the same basic orange
vector, orange line,

00:28:22.000 --> 00:28:28.000
but it has now traversed the
solution.

00:28:25.000 --> 00:28:31.000
We traverse it in the opposite
direction.

00:28:30.000 --> 00:28:36.000
Now, let's do the same thing
about dominance,

00:28:35.000 --> 00:28:41.000
as we did before.
Which term dominates as t goes

00:28:40.000 --> 00:28:46.000
to infinity?
This is the dominant term.

00:28:44.000 --> 00:28:50.000
Because, as t goes to infinity,
3t is much bigger than t.

00:28:51.000 --> 00:28:57.000
This one, on the other hand,
dominates as t goes to negative

00:28:57.000 --> 00:29:03.000
infinity.

00:29:05.000 --> 00:29:11.000
How now will the solutions look
like?

00:29:07.000 --> 00:29:13.000
Well, as t goes to infinity,
they follow the pink curve.

00:29:11.000 --> 00:29:17.000
Whereas, as t starts out from
negative infinity,

00:29:15.000 --> 00:29:21.000
they follow the orange curve.

00:29:28.000 --> 00:29:34.000
As t goes to infinity,
they become parallel to the

00:29:33.000 --> 00:29:39.000
pink curve, and as t goes to
negative infinity,

00:29:38.000 --> 00:29:44.000
they are very close to the
origin and are following the

00:29:44.000 --> 00:29:50.000
yellow curve.
This is pink and this is

00:29:48.000 --> 00:29:54.000
yellow.
They look like this.

00:30:03.000 --> 00:30:09.000
Notice the picture basically is
the same.

00:30:06.000 --> 00:30:12.000
It is the picture of a node.
All that has happened is the

00:30:11.000 --> 00:30:17.000
arrows are reversed.
And, therefore,

00:30:14.000 --> 00:30:20.000
this would be called a nodal
source.

00:30:17.000 --> 00:30:23.000
The word source and sink
correspond to what you learned

00:30:21.000 --> 00:30:27.000
in 18.02 and 8.02,
I hope, also,

00:30:24.000 --> 00:30:30.000
or you could call it a source
node.

00:30:27.000 --> 00:30:33.000
Both phrases are used,
depending on how you want to

00:30:31.000 --> 00:30:37.000
use it in a sentence.
And another word for this,

00:30:37.000 --> 00:30:43.000
this would be called unstable
because all of the solutions

00:30:41.000 --> 00:30:47.000
starting out from near the
origin ultimately end up

00:30:45.000 --> 00:30:51.000
infinitely far away from the
origin.

00:30:47.000 --> 00:30:53.000
This would be called stable.
In fact, it would be called

00:30:52.000 --> 00:30:58.000
asymptotically stable.
I don't like the word

00:30:55.000 --> 00:31:01.000
asymptotically,
but it has become standard in

00:30:58.000 --> 00:31:04.000
the literature.
And, more important,

00:31:02.000 --> 00:31:08.000
it is standard in your
textbook.

00:31:05.000 --> 00:31:11.000
And I don't like to fight with
a textbook.

00:31:08.000 --> 00:31:14.000
It just ends up confusing
everybody, including me.

00:31:12.000 --> 00:31:18.000
That is enough for nodes.
I would like to talk now about

00:31:16.000 --> 00:31:22.000
some of the other cases that can
occur because they lead to

00:31:21.000 --> 00:31:27.000
completely different pictures
that you should understand.

00:31:26.000 --> 00:31:32.000
Let's look at the case where
our governors behave a little

00:31:30.000 --> 00:31:36.000
more badly, a little more
combatively.

00:31:40.000 --> 00:31:46.000
It is x prime equals negative x
as before,

00:31:46.000 --> 00:31:52.000
but this time a firm response
by Massachusetts to any sign of

00:31:52.000 --> 00:31:58.000
increased activity by
stockpiling of advertising

00:31:58.000 --> 00:32:04.000
budgets.
Here let's say New Hampshire

00:32:03.000 --> 00:32:09.000
now is even worse.
Five times, quintuple or

00:32:08.000 --> 00:32:14.000
whatever increase Massachusetts
makes, of course they don't have

00:32:15.000 --> 00:32:21.000
an income tax,
but they will manage.

00:32:19.000 --> 00:32:25.000
Minus 3y as before.
Let's again calculate quickly

00:32:24.000 --> 00:32:30.000
what the characteristic equation
is.

00:32:30.000 --> 00:32:36.000
Our matrix is now negative 1,
3, 5 and negative 3.

00:32:34.000 --> 00:32:40.000
The characteristic equation now

00:32:37.000 --> 00:32:43.000
is lambda squared.
What is that?

00:32:40.000 --> 00:32:46.000
Again, plus 4 lambda.
But now the determinant is 3

00:32:44.000 --> 00:32:50.000
minus 15 is negative 12.

00:32:48.000 --> 00:32:54.000
And this, because I prepared
very carefully,

00:32:52.000 --> 00:32:58.000
all eigenvalues are integers.
And so this factors into lambda

00:32:57.000 --> 00:33:03.000
plus 6 times lambda minus 2,

00:33:01.000 --> 00:33:07.000
does it not?
Yes.

00:33:04.000 --> 00:33:10.000
6 lambda minus 2 is four
lambda.

00:33:07.000 --> 00:33:13.000
Good.
What do we have?

00:33:10.000 --> 00:33:16.000
Well, first of all we have our
eigenvalue lambda,

00:33:15.000 --> 00:33:21.000
negative 6.
And the eigenvector that goes

00:33:19.000 --> 00:33:25.000
with that is minus 1.
This is negative 1 minus

00:33:24.000 --> 00:33:30.000
negative 6 which makes,
shut your eyes,

00:33:28.000 --> 00:33:34.000
5.
We have 5a1 plus 3a2 is zero.

00:33:32.000 --> 00:33:38.000
And the other equation,

00:33:35.000 --> 00:33:41.000
I hope it comes out to be
something similar.

00:33:38.000 --> 00:33:44.000
I didn't check.
I am hoping this is right.

00:33:42.000 --> 00:33:48.000
The eigenvector is,
okay, you have been taught to

00:33:46.000 --> 00:33:52.000
always make one of the 1,
forget about that.

00:33:49.000 --> 00:33:55.000
Just pick numbers that make it
come out right.

00:33:53.000 --> 00:33:59.000
I am going to make this one 3,
and then I will make this one

00:33:57.000 --> 00:34:03.000
negative 5.
As I say, I have a policy of

00:34:02.000 --> 00:34:08.000
integers only.
I am a number theorist at

00:34:06.000 --> 00:34:12.000
heart.
That is how I started out life

00:34:09.000 --> 00:34:15.000
anyway.
There we have data from which

00:34:12.000 --> 00:34:18.000
we can make one solution.
How about the other one?

00:34:17.000 --> 00:34:23.000
The other one will correspond
to the eigenvalue lambda equals

00:34:23.000 --> 00:34:29.000
2.
This time the equation is

00:34:25.000 --> 00:34:31.000
negative 1 minus 2 is negative
3.

00:34:30.000 --> 00:34:36.000
It is minus 3a1 plus 3a2 is
zero.

00:34:34.000 --> 00:34:40.000
And now the eigenvector is (1,
1).

00:34:37.000 --> 00:34:43.000
Now we are ready to draw
pictures.

00:34:40.000 --> 00:34:46.000
We are going to make this
similar analysis,

00:34:44.000 --> 00:34:50.000
but it will go faster now
because you have already had the

00:34:49.000 --> 00:34:55.000
experience of that.
First of all,

00:34:52.000 --> 00:34:58.000
what is our general solution?
It is going to be c1 times 3,

00:34:57.000 --> 00:35:03.000
negative 5 e to the minus 6t.

00:35:02.000 --> 00:35:08.000
And then the other normal mode

00:35:06.000 --> 00:35:12.000
times an arbitrary constant will
be 1, 1 times e to the 2t.

00:35:18.000 --> 00:35:24.000
I am going to use the same
strategy.

00:35:20.000 --> 00:35:26.000
We have our two normal modes
here, eigenvalue,

00:35:24.000 --> 00:35:30.000
eigenvector solutions from
which, by adjusting these

00:35:27.000 --> 00:35:33.000
constants, we can get our four
basic solutions.

00:35:32.000 --> 00:35:38.000
Those are going to look like,
let's draw a picture here.

00:35:37.000 --> 00:35:43.000
Again, I will color-code them.
Let's use pink again.

00:35:42.000 --> 00:35:48.000
The pink solution now starts at
3, negative 5.

00:35:47.000 --> 00:35:53.000
That is where it is when t is

00:35:50.000 --> 00:35:56.000
zero.
And, because of the coefficient

00:35:54.000 --> 00:36:00.000
minus 6 up there,
it is coming into the origin

00:35:58.000 --> 00:36:04.000
and looks like that.
And its mirror image,

00:36:03.000 --> 00:36:09.000
of course, does the same thing.
That is when c1 is negative

00:36:08.000 --> 00:36:14.000
one.
How about the orange guy?

00:36:10.000 --> 00:36:16.000
Well, when t is equal to zero,
it is at 1, 1.

00:36:14.000 --> 00:36:20.000
But what is it doing after

00:36:16.000 --> 00:36:22.000
that?
As t increases,

00:36:18.000 --> 00:36:24.000
it is getting further away from
the origin because the sign here

00:36:22.000 --> 00:36:28.000
is positive.
e to the 2t is

00:36:25.000 --> 00:36:31.000
increasing, it is not decreasing
anymore, so this guy is going

00:36:30.000 --> 00:36:36.000
out.
And its mirror image on the

00:36:35.000 --> 00:36:41.000
other side is doing the same
thing.

00:36:40.000 --> 00:36:46.000
Now all we have to do is fill
in the picture.

00:36:46.000 --> 00:36:52.000
Well, you fill it in by
continuity.

00:36:51.000 --> 00:36:57.000
Your nearby trajectories must
be doing what similar thing?

00:37:00.000 --> 00:37:06.000
If I start out very near the
pink guy, I should stay near the

00:37:04.000 --> 00:37:10.000
pink guy.
But as I get near the origin,

00:37:07.000 --> 00:37:13.000
I am also approaching the
orange guy.

00:37:09.000 --> 00:37:15.000
Well, there is no other
possibility other than that.

00:37:13.000 --> 00:37:19.000
If you are further away you
start turning a little sooner.

00:37:17.000 --> 00:37:23.000
I am just using an argument
from continuity to say the

00:37:21.000 --> 00:37:27.000
picture must be roughly filled
out this way.

00:37:24.000 --> 00:37:30.000
Maybe not exactly.
In fact, there are fine points.

00:37:29.000 --> 00:37:35.000
And I am going to ask you to do
one of them on Friday for the

00:37:32.000 --> 00:37:38.000
new problem set,
even before the exam,

00:37:35.000 --> 00:37:41.000
God forbid.
But I want you to get a little

00:37:37.000 --> 00:37:43.000
more experience working with
that linear phase portrait

00:37:41.000 --> 00:37:47.000
visual because it is,
I think, one of the best ones

00:37:44.000 --> 00:37:50.000
this semester.
You can learn a lot from it.

00:37:47.000 --> 00:37:53.000
Anyway, you are not done with
it, but I hope you have at least

00:37:51.000 --> 00:37:57.000
looked at it by now.
That is what the picture looks

00:37:54.000 --> 00:38:00.000
like.
First of all,

00:37:55.000 --> 00:38:01.000
what are we going to name this?
In other words,

00:38:00.000 --> 00:38:06.000
forget about the arrows.
If you just look at the general

00:38:05.000 --> 00:38:11.000
way those lines go,
where have you seen this

00:38:08.000 --> 00:38:14.000
before?
You saw this in 18.02.

00:38:11.000 --> 00:38:17.000
What was the topic?
You were plotting contour

00:38:15.000 --> 00:38:21.000
curves of functions,
were you not?

00:38:18.000 --> 00:38:24.000
What did you call contours
curves that formed that pattern?

00:38:23.000 --> 00:38:29.000
A saddle point.
You called this a saddle point

00:38:26.000 --> 00:38:32.000
because it was like the center
of a saddle.

00:38:32.000 --> 00:38:38.000
It is like a mountain pass.
Here you are going up the

00:38:35.000 --> 00:38:41.000
mountain, say,
and here you are going down,

00:38:37.000 --> 00:38:43.000
the way the contour line is
going down.

00:38:40.000 --> 00:38:46.000
And this is sort of a min and
max point.

00:38:42.000 --> 00:38:48.000
A maximum if you go in that
direction and a minimum if you

00:38:46.000 --> 00:38:52.000
go in that direction,
say.

00:38:48.000 --> 00:38:54.000
Without the arrows on it,
it is like a saddle point.

00:38:51.000 --> 00:38:57.000
And so the same word is used
here.

00:38:53.000 --> 00:38:59.000
It is called the saddle.
You don't say point in the same

00:38:56.000 --> 00:39:02.000
way you don't say a nodal point.
It is the whole picture,

00:39:01.000 --> 00:39:07.000
as it were, that is the saddle.
It is a saddle.

00:39:05.000 --> 00:39:11.000
There is the saddle.
This is where you sit.

00:39:08.000 --> 00:39:14.000
Now, should I call it a source
or a sink?

00:39:12.000 --> 00:39:18.000
I cannot call it either because
it is a sink along these lines,

00:39:16.000 --> 00:39:22.000
it is a source along those
lines and along the others,

00:39:21.000 --> 00:39:27.000
it starts out looking like a
sink and then turns around and

00:39:25.000 --> 00:39:31.000
starts acting like a source.
The word source and sink are

00:39:31.000 --> 00:39:37.000
not used for saddle.
The only word that is used is

00:39:34.000 --> 00:39:40.000
unstable because definitely it
is unstable.

00:39:38.000 --> 00:39:44.000
If you start off exactly on the
pink lines you do end up at the

00:39:42.000 --> 00:39:48.000
origin, but if you start
anywhere else ever so close to a

00:39:47.000 --> 00:39:53.000
pink line you think you are
going to the origin,

00:39:50.000 --> 00:39:56.000
but then at the last minute you
are zooming off out to infinity

00:39:55.000 --> 00:40:01.000
again.
This is a typical example of

00:39:57.000 --> 00:40:03.000
instability.
Only if you do the

00:40:01.000 --> 00:40:07.000
mathematically possible,
but physically impossible thing

00:40:06.000 --> 00:40:12.000
of starting out exactly on the
pink line, only then will you

00:40:11.000 --> 00:40:17.000
get to the origin.
If you start out anywhere else,

00:40:15.000 --> 00:40:21.000
make the slightest error in
measure and get off the pink

00:40:20.000 --> 00:40:26.000
line, you end off at infinity.
What is the effect with our

00:40:25.000 --> 00:40:31.000
war-like governors fighting for
the tourist trade willing to

00:40:30.000 --> 00:40:36.000
spend any amounts of money to
match and overmatch what their

00:40:35.000 --> 00:40:41.000
competitor in the nearby state
is spending?

00:40:41.000 --> 00:40:47.000
The answer is,
they all lose.

00:40:43.000 --> 00:40:49.000
Since it is mostly this section
of the diagram that makes sense,

00:40:48.000 --> 00:40:54.000
what happens is they end up all
spending an infinity of dollars

00:40:53.000 --> 00:40:59.000
and nobody gets any more
tourists than anybody else.

00:40:58.000 --> 00:41:04.000
So this is a model of what not
to do.

00:41:02.000 --> 00:41:08.000
I have one more model to show
you.

00:41:05.000 --> 00:41:11.000
Maybe we better start over at
this board here.

00:41:11.000 --> 00:41:17.000
Massachusetts on top.
New Hampshire on the bottom.

00:41:17.000 --> 00:41:23.000
x prime is going to be,
that is Massachusetts,

00:41:23.000 --> 00:41:29.000
I guess as before.
Let me get the numbers right.

00:41:45.000 --> 00:41:51.000
Leave that out for a moment.
y prime is 2x minus 3y.

00:41:50.000 --> 00:41:56.000
New Hampshire behaves normally.

00:41:54.000 --> 00:42:00.000
It is ready to respond to
anything Massachusetts can put

00:41:59.000 --> 00:42:05.000
out.
But by itself,

00:42:01.000 --> 00:42:07.000
it really wants to bring its
budget to normal.

00:42:05.000 --> 00:42:11.000
Now, Massachusetts,
we have a Mormon governor now,

00:42:09.000 --> 00:42:15.000
I guess.
Imagine instead we have a

00:42:12.000 --> 00:42:18.000
Buddhist governor.
A Buddhist governor reacts as

00:42:16.000 --> 00:42:22.000
follows, minus y.
What does that mean?

00:42:20.000 --> 00:42:26.000
It means that when he sees New
Hampshire increasing the budget,

00:42:25.000 --> 00:42:31.000
his reaction is,
we will lower ours.

00:42:30.000 --> 00:42:36.000
We will show them love.
It looks suicidal,

00:42:34.000 --> 00:42:40.000
but what actually happens?
Well, our little program is

00:42:39.000 --> 00:42:45.000
over.
Our matrix a is negative 1,

00:42:42.000 --> 00:42:48.000
negative 1, 2,
negative 3.

00:42:46.000 --> 00:42:52.000
The characteristic equations is

00:42:50.000 --> 00:42:56.000
lambda squared plus 4 lambda.

00:42:55.000 --> 00:43:01.000
And now what is the other term?
3 minus negative 2 makes 5.

00:43:02.000 --> 00:43:08.000
This is not going to factor
because I tried it out and I

00:43:07.000 --> 00:43:13.000
know it is not going to factor.
We are going to get lambda

00:43:13.000 --> 00:43:19.000
equals, we will just use the
quadratic formula,

00:43:17.000 --> 00:43:23.000
negative 4 plus or minus the
square root of 16 minus 4 times

00:43:23.000 --> 00:43:29.000
5, that is 16 minus 20 or
negative 4 all divided by 2,

00:43:28.000 --> 00:43:34.000
which makes minus 2,
pull out the 4,

00:43:31.000 --> 00:43:37.000
that makes it a 2,
cancels this 2,

00:43:35.000 --> 00:43:41.000
minus 1 inside.
It is minus 2 plus or minus i.

00:43:40.000 --> 00:43:46.000
Complex solutions.

00:43:44.000 --> 00:43:50.000
What are we doing to do about
that?

00:43:47.000 --> 00:43:53.000
Well, you should rejoice when
you get this case and are asked

00:43:53.000 --> 00:43:59.000
to sketch it because,
even if you calculate the

00:43:58.000 --> 00:44:04.000
complex eigenvector and from
that take its real and imaginary

00:44:04.000 --> 00:44:10.000
parts of the complex solution,
in fact, you will not be able

00:44:10.000 --> 00:44:16.000
easily to sketch the answer
anyway.

00:44:15.000 --> 00:44:21.000
But let me show you what sort
of thing you can get and then I

00:44:18.000 --> 00:44:24.000
am going to wave my hands and
argue a little bit to try to

00:44:21.000 --> 00:44:27.000
indicate what it is that the
solution actually looks like.

00:44:24.000 --> 00:44:30.000
You are going to get something
that looks like --

00:44:28.000 --> 00:44:34.000
A typical real solution is
going to look like this.

00:44:31.000 --> 00:44:37.000
This is going to produce e to
the minus 2t times e

00:44:36.000 --> 00:44:42.000
to the i t.
e to the minus 2 plus i all

00:44:40.000 --> 00:44:46.000
times t.
This will be our exponential

00:44:44.000 --> 00:44:50.000
factor which is shrinking in
amplitude.

00:44:47.000 --> 00:44:53.000
This is going to give me sines
and cosines.

00:44:50.000 --> 00:44:56.000
When I separate out the
eigenvector into its real and

00:44:54.000 --> 00:45:00.000
imaginary parts,
it is going to look something

00:44:57.000 --> 00:45:03.000
like this.
a1, a2 times cosine t,

00:45:02.000 --> 00:45:08.000
that is from the e to the it

00:45:05.000 --> 00:45:11.000
part.
Then there will be a sine term.

00:45:08.000 --> 00:45:14.000
And all that is going to be
multiplied by the exponential

00:45:12.000 --> 00:45:18.000
factor e to the negative 2t.

00:45:22.000 --> 00:45:28.000
That is just one normal mode.
It is going to be c1 times this

00:45:28.000 --> 00:45:34.000
plus c2 times something similar.
It doesn't matter exactly what

00:45:34.000 --> 00:45:40.000
it is because they are all going
to look the same.

00:45:37.000 --> 00:45:43.000
Namely, this is a shrinking
amplitude.

00:45:40.000 --> 00:45:46.000
I am not going to worry about
that.

00:45:42.000 --> 00:45:48.000
My real question is,
what does this look like?

00:45:45.000 --> 00:45:51.000
In other words,
as a pair of parametric

00:45:48.000 --> 00:45:54.000
equations, if x is equal to a1
cosine t plus b1 sine t

00:45:52.000 --> 00:45:58.000
and y is a2
cosine plus b2 sine,

00:45:56.000 --> 00:46:02.000
what does it look like?

00:46:01.000 --> 00:46:07.000
Well, what are its
characteristics?

00:46:03.000 --> 00:46:09.000
In the first place,
as a curve this part of it is

00:46:08.000 --> 00:46:14.000
bounded.
It stays within some large box

00:46:11.000 --> 00:46:17.000
because cosine and sine never
get bigger than one and never

00:46:16.000 --> 00:46:22.000
get smaller than minus one.
It is periodic.

00:46:20.000 --> 00:46:26.000
As t increases to t plus 2pi,

00:46:24.000 --> 00:46:30.000
it comes back to exactly the
same point it was at before.

00:46:35.000 --> 00:46:41.000
We have a curve that is
repeating itself periodically,

00:46:38.000 --> 00:46:44.000
it does not go off to infinity.
And here is where I am waving

00:46:42.000 --> 00:46:48.000
my hands.
It satisfies an equation.

00:46:44.000 --> 00:46:50.000
Those of you who like to fool
around with mathematics a little

00:46:49.000 --> 00:46:55.000
bit, it is not difficult to show
this, but it satisfies an

00:46:52.000 --> 00:46:58.000
equation of the form A x squared
plus B y squared plus C xy

00:46:56.000 --> 00:47:02.000
equals D.

00:47:00.000 --> 00:47:06.000
All you have to do is figure
out what the coefficients A,

00:47:03.000 --> 00:47:09.000
B, C and D should be.
And the way to do it is,

00:47:06.000 --> 00:47:12.000
if you calculate the square of
x you are going to get cosine

00:47:10.000 --> 00:47:16.000
squared, sine squared and a
cosine sine term.

00:47:13.000 --> 00:47:19.000
You are going to get those same
three terms here and the same

00:47:17.000 --> 00:47:23.000
three terms here.
You just use undetermined

00:47:20.000 --> 00:47:26.000
coefficients,
set up a system of simultaneous

00:47:23.000 --> 00:47:29.000
equations and you will be able
to find the A,

00:47:26.000 --> 00:47:32.000
B, C and D that work.
I am looking for a curve that

00:47:31.000 --> 00:47:37.000
is bounded, keeps repeating its
values and that satisfies a

00:47:35.000 --> 00:47:41.000
quadratic equation which looks
like this.

00:47:38.000 --> 00:47:44.000
Well, an earlier generation
would know from high school,

00:47:42.000 --> 00:47:48.000
these curves are all conic
sections.

00:47:45.000 --> 00:47:51.000
The only curves that satisfy
equations like that are

00:47:48.000 --> 00:47:54.000
hyperbola, parabolas,
the conic sections in other

00:47:52.000 --> 00:47:58.000
words, and ellipses.
Circles are a special kind of

00:47:56.000 --> 00:48:02.000
ellipses.
There is a degenerate case.

00:48:00.000 --> 00:48:06.000
A pair of lines which can be
considered a degenerate

00:48:04.000 --> 00:48:10.000
hyperbola, if you want.
It is as much a hyperbola as a

00:48:08.000 --> 00:48:14.000
circle, as an ellipse say.
Which of these is it?

00:48:11.000 --> 00:48:17.000
Well, it must be those guys.
Those are the only guys that

00:48:16.000 --> 00:48:22.000
stay bounded and repeat
themselves periodically.

00:48:20.000 --> 00:48:26.000
The other guys don't do that.
These are ellipses.

00:48:23.000 --> 00:48:29.000
And, therefore,
what do they look like?

00:48:28.000 --> 00:48:34.000
Well, they must look like an
ellipse that is trying to be an

00:48:32.000 --> 00:48:38.000
ellipse, but each time it goes
around the point is pulled a

00:48:37.000 --> 00:48:43.000
little closer to the origin.
It must be doing this,

00:48:41.000 --> 00:48:47.000
in other words.
And such a point is called a

00:48:44.000 --> 00:48:50.000
spiral sink.
Again sink because,

00:48:47.000 --> 00:48:53.000
no matter where you start,
you will get a curve that

00:48:51.000 --> 00:48:57.000
spirals into the origin.
Spiral is self-explanatory.

00:48:55.000 --> 00:49:01.000
And the one thing I haven't
told you that you must read is

00:49:00.000 --> 00:49:06.000
how do you know that it goes
around counterclockwise and not

00:49:04.000 --> 00:49:10.000
clockwise?
Read clockwise or

00:49:08.000 --> 00:49:14.000
counterclockwise.
I will give you the answer in

00:49:12.000 --> 00:49:18.000
30 seconds, not for this
particular curve.

00:49:16.000 --> 00:49:22.000
That you will have to
calculate.

00:49:19.000 --> 00:49:25.000
All you have to do is put in
somewhere.

00:49:23.000 --> 00:49:29.000
Let's say at the point (1,
0), a single vector from the

00:49:28.000 --> 00:49:34.000
velocity field.
In other words,

00:49:32.000 --> 00:49:38.000
at the point (1,
0), when x is 1 and y is 0 our

00:49:37.000 --> 00:49:43.000
vector is minus 1, 2,

00:49:40.000 --> 00:49:46.000
which is the vector minus 1,
2, it goes like this.

00:49:46.000 --> 00:49:52.000
Therefore, the motion must be
counterclockwise.

00:49:51.000 --> 00:49:57.000
And, by the way,
what is the effect of having a

00:49:56.000 --> 00:50:02.000
Buddhist governor?
Peace.

00:50:00.000 --> 00:50:06.000
Everything spirals into the
origin and everybody is left

00:50:05.000 --> 00:50:11.000
with the same advertising budget
they always had.

00:50:10.000 --> 00:50:16.000
Thanks.