We finish off our discussion of stability
and of long-time behavior by considering
what happens with second order
ordinary differential equations.
That is, systems that look like the
second derivative of a vector
is a matrix times the vector.
Now, we've seen that whenever we
have a system like that,
every eigenvalue of A and every
eigenvector gives you two terms
in the expansion of x of t. If you have
a positive eigenvalue, then we're used
to saying that the coefficient of the
eigenvector is some constant
times the cosh, plus some constant
times the sinh. We could just as well
express the coefficient as some
constant times e to the kappa t,
plus some constant times e to the
minus kappa t,
where kappa is the square root of
the eigenvalue. Now in the long run,
unless this happens to be zero,
this term is going to grow.
This term is going to shrink, but this
one's going to grow.
And so the whole thing grows like e to
the kappa t, and it's unstable.
So whenever you have a positive
eigenvalue you say it's unstable
with a growth going as e to the square
root of lambda times t.
This is what you have whenever you
have a ball at the top of a hill.
You bump it a little bit and it falls off
and falls off faster, and faster,
and faster. Now there are trajectories
that crawl up the hill and get closer,
and closer, and closer to the top,
and closer to the top,
and closer to the top, that correspond
to that. But generically,
that doesn't happen. That's saying that
the first coefficient was exactly zero.
It's never exactly zero. It's always
slightly positive or slightly negative.
And so that ball reaches the top and
falls down the other side.
Whatever it is, it's unstable, it runs
away faster, and faster, and faster.
If you have an eigenvalue that's
negative, then we define a frequency
to be the square root of minus the
eigenvalue. So if this is negative four,
the frequency will be square root
of four, or two.
In other words, the square root of
the eigenvalue is an imaginary number,
i omega.
We get the coefficient of the eigenvector
is something times cosine omega t,
plus something times sine omega t.
Or we could just as well express it
in terms of exponentials. e to the i
omega t, that is e to the square root
of lambda t, and e to the minus i omega
t, that is e to the minus square root
of lambda t. Now whether you think of it
as cosines and sines
or as complex exponentials, these don't
go anywhere. You wait a million years
and this term is exactly as big as it ever
was, all that changed was its phase.
This term is exactly as big as it
ever was. This goes up and down,
and up and down, and up and down, but
it doesn't grow or shrink in the long run
and neither does this. These are all
neutral modes.
That describes what happens if you 
have a ball rolling at the bottom of a hill.
If you give it a little bit of kick,
it goes back and forth,
and back and forth, and back and forth,
forever. It never escapes the well,
but it never settles down either. At least
not if there's - it'll settle down
if you had friction, but these equations
are describing a system with no friction,
and it would just go forever. That's what
we call neutral, it doesn't grow,
it doesn't shrink, it just keeps on going
back and forth, and back and forth.
And the last case is if the eigenvalue
is zero. This would describe a ball
rolling on a flat track. It rolls and it
keeps rolling, rolling, rolling, rolling,
and it doesn't speed up and it doesn't
slow down. And its position is a constant
plus a constant times t. We call this
borderline, but it's not borderline
between stable and unstable,
it's borderline between neutral
and unstable. It actually does grow,
it just doesn't grow exponentially.
It grows merely as t to the first power.
So bottom line, if any of the eigenvalues
are positive, the system's
going to be unstable
because if any of the eigenvalues
are positive, you're going to have one
term in your expansion that goes
as e to a positive number times t.
And the dominant mode is
whichever eigenvalue has the greatest
positive - which is the most positive.
But if all the eigenvalues are negative,
then there is no dominant mode
because the system is neutral.
None of the modes grow.
None of the modes shrink. They all just
oscillate back and forth,
and back and forth, and there's no one
mode that you would call dominant.
Finally, I showed you a chart in the
previous ones where you had which part
of the complex plane was stable,
which was unstable, which was neutral.
For our second order systems, the matrices
almost always have real eigenvalues.
So I'm not going to draw the whole
complex plane, I'm just going to draw
the real line. The eigenvalues that are
bigger than zero are unstable.
The eigenvalues that are less than zero
are neutrally stable,
and zero is borderline.
So that's the story for second order
differential equations.
