In the last video, we talked about
stability and long-time behavior
for difference equations. Now we are
gonna do it for first order ordinary
differential equations, systems of
differential equations.
The overall picture is very much 
the same but some of the details
are different.
So first of all, when you have a system
of differential equations, you wind up
with behaviors like e ^ λt. So suppose
you had a vector whose entries were,
.01 e^2t, e ^ (1+i)t + e ^ (1-i)t,
and the difference of the complex
exponential is divided by 2i and then
1,000,000 e^-3t, and I ask you,
what does this do in the long run?
Well you notice that this term
grows and these terms grow.
These grow and oscillate but they grow.
And this term shrinks. So it you 
expand it out like this, you'd say,
"hmm, I've got a term that grows,
a term that grows, a term that grows."
The sum of the complex exponentials
and the differences wind up turning into
exponential times cosine and exponential
times sine and then a term that shrinks.
And so in the long run, you wind up
with something that behaves like
the term that grows the fastest. 
e^2t grows faster than e^t.
This term shrinks away.
This term shrinks.
These terms grow slowly.
And this term grows quickly.
So in the long run, this term is gonna
be the biggest term and it's gonna-
and eventually, the vector's gonna be
pointing more or less in the e_1 direction
and it's gonna be growing like e^2t.
So in general, when you've got
a system of ordinary differential 
equations, dx/dt equals a matrix times x.
We know that the solution is linear 
combination of terms, but the terms
are e to the eigenvalue times t,
times the eigenvector.
And we say hey, if the eigenvalue,
if we have an eigenvalue, call it 'a'
and it's positive, then e^at grows.
And we call that unstable.
If the eigenvalue is negative, then 
e^at shrinks, and we call that stable.
And if the eigenvalue is zero, then 
e^at is just 1, and we call that
neutral or borderline. But that leaving
something out. The thing that's left out
is complex eigenvalues. Eigenvalues
don't have to be real numbers.
They can have real and imaginary parts.
So if you have an eigenvalue that's
a + bi, then we have to look at what is
e^λt, how does that behave?
And we split it up as e^at and e^bit.
And e^bit is cos(bt) +i sin(bt).
This oscillates. The real part goes up
and down and up and down
and up and down and the imaginary
part goes up and down and up and down
and up and down. The whole thing
oscillates but the whole thing grows
like e^at. So it grows if a is positive.
It shrinks if a is negative.
It's borderline if a is 0.
In other words, if the real part of
the eigenvalue is positive, it will grow 
no matter what the imaginary part is.
If the real part is negative, it will shrink
no matter what the imaginary part is.
The real part is 0, it will be neutral
no matter what the imaginary part is.
So now let's go back and look at our
original picture. We had these terms
that went as e^(1+i)t and e^(1-i)t.
e^(1+i)t is e^1t, and then you have
an e^it that's cos + i sin.
e^-i is cos - i sin,
and they add up to give you 2 cosine.
And likewise here you have things
that grow as e^1t and e^1t,
and they happen to cancel and give you
exactly sine. But the point is,
that this 1, the real part of
the eigenvalue tells you how fast it 
grows. This is less than 2, which
is the real part of that eigenvalue,
so this term grows faster than
these terms. The dominant eigenvalue
is always the eigenvalue with the greatest
real part. 2 is bigger than 1.
And by the way I am not saying
the greatest in absolute value.
2 is bigger than -5.
-5 would be stable. You would have
something that shrinks.
2 grows so the eigenvalue with
the greatest real part is the dominant
eigenvalue and that gives you the term
that grows the fastest or shrinks
the slowest and it's what your system
will look like in the long run.
So here is a picture in the complex plane
of what's stable and what's unstable.
The whole right half plane is unstable.
Anything with a positive real part
is unstable. Anything with a negative
real part, the left half plane, is stable.
The neutral ones are the ones
with 0 real part. In other words,
exactly the imaginary axis.
3i is neutral. -5i is neutral.
0 is neutral. And just as with difference
equations, for a system to be stable,
you have to have all of the modes
stable. In other words, all of
the eigenvalues have to be in the left
hand side. So this picture is where
you have the biggest difference between
differential equations and difference
equations. For difference equations,
we have inside the unit circle,
outside the unit circle,
on the unit circle.
For differential equations, you have
the right half plane, the left half plane
and the imaginary axis.
