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GILBERT STRANG: OK?

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Now, finally, a
nonlinear equation.

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Growth, but it's-- the growth
is cut off by competition--

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slowed down my competition.

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Let me show you the equation.

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In a way, it's the
simplest nonlinear equation

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I could think of.

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We have the usual
dy/dt equal ay.

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That would give
exponential growth.

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The growth would go on forever.

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But then, as y
gets large-- if we

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think of y as the population,
so the growth is exponential

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when the birth rates higher than
the death rate, and we grow.

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We have-- and that's
what's happening.

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But our world population is
not growing pure exponentially,

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forever.

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And this is a simple
term, a simple model not,

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very accurate, but it's
the right place to start.

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For competition, when
you have too many people,

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somehow the
multiplying y times y

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gives you a number of
interactions of all the y

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people with themselves.

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And those interactions--
that competition slows down

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the growth, and so the
coefficient there is negative.

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We want to solve that non-linear
equation and learn from it.

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And it's called the
logistic equation.

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That's-- it's got to
be a famous example.

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And it has a neat trick that
allows you to solve it easily.

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Let me show you that trick.

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The trick is to let z--
bring in a new z as 1/y.

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Then, if I write
the equation for z,

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it will turn out to be linear.

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So you see, I'm always
hoping, and here succeeding,

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to get back to a
simple linear equation.

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And this device happens
to work for this problem.

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We need, in the end,
a more systematic way

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that would work
for other problems,

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like there could be
some constant term

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there from harvesting, from
the changing the equation.

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Let me go with this one.

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So what's the equation
for z then, dz/dt.

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The derivative of that,
that's y to the minus 1

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is minus 1 y to the minus
2 times dy/dt, right?

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The derivative of y
the minus 1 is minus 1,

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y to the minus 2 times
the derivative of dy/dt.

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But now dy/dt is ay
minus by squared.

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And what do I have?

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I've got a, with a
minus, ay over y squared.

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That will be z.

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So that's minus az.

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Minus minus giving me a plus b.

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y squared over y squared
is just a constant.

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Look at the equation for z now.

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Couldn't be better.

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dz/dt is minus az
and plus a constant.

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And we know the solution
to that equation.

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So I can write down the
solution to that equation.

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And I will.

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Then, I just take y to
be 1 over that solution.

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Maybe, I'll just write
y straight ahead.

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I'll just write the
answer y, which is 1/z.

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And z is the solution
to this equation.

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And let me see.

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So I'm going to
have an exponential,

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but it's going to be-- the
exponential e to the minus at

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is going to show up in z.

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And they'll be more there.

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It will show up in z.

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And then when I do
1/z to get back to y,

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that exponential is going
into the denominator.

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That's what's new.

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That's what's new.

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Let me get the
whole thing correct.

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I have a a.

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Otherwise, these are constants
d here and b goes there.

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And I just have to tell
you what the d is there.

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d is going to involve the
initial value, y of 0.

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At time 0, we start
somewhere, y of 0.

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And we'll begin to grow.

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We'll grow.

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But with the slowing down
term keeping us back.

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And, in the end, it
can't grow forever

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because that term becomes
bigger than that one.

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If y went on and on increasing,
then it can't happen.

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So it really is--
it can only increase

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while this is
positive, and that's

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only positive below that line.

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Here's what the curve does.

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It goes upward, but
then it has to-- it

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can't cross that line
because above that line,

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for big Y-- above
that line for big Y

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this negative term means
that the slope is negative.

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If I started with a big
number, I'd have to come down.

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But can I just
draw that picture?

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But it's really between
here-- between these lines

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that I'm thinking the
world population is.

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So it started way back at
some time minus infinity.

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And it-- whatever.

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It came up to time 0.

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This is t equals 0,
starting population.

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And, of course, t equals
0 could be the year 2000.

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We can move the time
axis as we want.

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So t equals 0 might
be the year 2000.

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Our population is
still increasing,

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but it won't go on forever.

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And it will slow down and
approach this limit a/b--

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approaches a/b.

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So that number a/b is totally
special for this equation.

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And let me think
about that part.

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And we give a name
to this curve.

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It's often called
a logistic curve.

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That's OK.

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But I like the name S curve.

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It's an S curve.

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In some way-- or
a Sigmoidal curve.

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In some way, it looks a bit like
an S, a very stretched out S.

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So that-- and you see
why-- do you-- just

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takes like-- look here.

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As t goes to infinity,
this becomes 0.

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When t gets large, this
exponential is gone.

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And I just have a
or b at the limit.

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When t is minus infinity,
way back at starting time--

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when t is minus infinity,
that's a giant number.

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This is-- I'm dividing
by a giant number.

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I'm at 0.

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So way back at time minus
infinity, it started from 0.

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Those two possibilities ,
y equals 0 and y equal a/b,

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are the key things to
see for this equation.

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So I'm going to
take a few minutes

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to see where do those come in.

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Those are steady states.

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Let me use that word and
show you what I mean.

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Steady states, critical
points, for my equation dy/dt.

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Let me write the
logistic equation again.

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A steady state is a value of
y where the derivative is 0.

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Nothing happens.

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It just sits there.

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So if that is 0--
so a steady state--

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let me use those capital
Y for a steady state.

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Those are just numbers.

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They are the numbers where this
is 0, ay minus by squared is 0.

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You see that if
I start out at y,

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at capital Y, a steady
state, that is 0.

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The derivative is 0.

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I stay there.

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And there are two possibilities.

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If I start-- this gives me
two possibilities, right?

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What are the solutions
to this equation?

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This gives me y equals 0
is one solution clearly.

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Or y equals-- let's see.

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If this equals that,
I could cancel a y.

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y would be a/b, exactly
the two in the picture.

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If I start at 0, I stay at 0.

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No population.

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Nothing can happen.

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But if I move a little away,
if I get two people, we grow.

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We grow exponentially
for awhile,

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but then the by squared term
takes it-- gets in there,

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slows us down.

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And we slow down, and
we reach this a/b.

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If we had started at a/b,
we would have stayed at a/b.

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So that S curve
picture is really

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a nice graph of the solution.

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Here is the solution at
steady state, going nowhere.

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There is a solution at
steady state with population,

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competition, and cooperation are
balanced, and nothing happens.

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Otherwise, whenever we have S
curves, then they don't cross.

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Depending where we're--
depending on the initial value,

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we'll get beyond one or
other of these curves.

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And they all go up to
that carrying capacity,

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that limiting population, a/b.

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So the final take home
message about this equation

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is, as an example of
equations with steady states,

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this steady state
is called unstable

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because if we start near
y equals 0, we take off.

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If we start near y equals
0, we start growing.

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We leave that steady state.

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This steady state
is called stable

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because if we're near a/b, we
get closer and closer to a/b.

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I'll do the example.

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This will be one example
of a general problem.

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dy/dt is some function of y.

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It's a right-hand side
that depends on y.

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That will be our next
video, to understand

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those equations dy/dt
is f of y and notice

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that they're not linear.

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But we can solve
for steady states.

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A steady state will
be when f at y is 0.

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Then, nothing happens.

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But is that steady state stable?

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Or is it unstable?

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That's the crucial thing
for all applications.

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Are we settling down to
something acceptable?

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Or are we blowing up to
something impossible?

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That's the next video.

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Thank you.