Our next topic is the stability
and long-term behavior of
evolution systems. So, we're gonna
start with difference equations,
and in a subsequent lecture we're
gonna talk about first-order
differential equations, and then
second-order differential equations.
So, suppose I gave you a vector, you
know, a vector is a function of n,
and it was a million times .6 to the n,
and 4, and .0000001 times 3 to the n,
and I ask you, how does that behave
in the long run? Well, you might,
you know write it as a sum of three terms,
and you'd look at what each term does in
the long run. This term, it starts off as
a million, but every turn it gets cut
down to 60% of what it had been the
previous term. .6 to the n goes to 0,
so this whole term shrinks.
Now, this term, 4e2, it just stays
there. It doesn't shrink, it doesn't grow.
And the last term grows. Now, it
starts off real small, but it doesn't
matter how small it started off, it's
gonna keep growing and growing and
growing and growing and growing.
In the long run, this term is gonna
be much bigger than the other two
terms put together.
The fact that the coefficient was small
is irrelevant, the fact that the power
that- the number that you're multiplying-
the number that you're taking to the nth
power is bigger than 1, that matters.
3 to the n grows, 1 to the n doesn't
do anything, .6 to the n shrinks.
Now, this kind of behavior is typical
of what you'd get from a difference
equation, so if you start off with the
difference equation where x today
is a matrix times x yesterday, we know
how to solve that, and our general
solution is gonna be, you diagonalize
A, you find the eigenvalues and
eigenvectors, and it's gonna be a linear
combination of the eigenvectors,
where the coefficients go as the
eigenvalue to the nth power.
So whenever the eigenvalue is smaller
than 1, you take powers of a number
less than 1, and they get smaller and
smaller. It doesn't matter what the
coefficient was, they shrink, and our
term for things that shrink, is we're
gonna call them stable, and any term
that has an eigenvalue bigger than
1 is gonna grow, you know, you take
3 to the n, or -2 to the n, or 5 to the
n, these get bigger and bigger,
and those are unstable,
and if you have a term who's eigenvalue
is size 1, then it sticks around,
it doesn't grow, it doesn't shrink,
and we call them neutral,
or borderline. So every term in
this expansion we're gonna call
a mode, and we have stable modes
that shrink, unstable modes that grow,
and neutral modes that stick around.
Now, of all the modes, there's gonna
be one of them that has the biggest
eigenvalue. Whatever the biggest
eigenvalue is, that term is going to
either grow faster than all of the
others, or shrink slower than all
of the others, and so it will eventually
be much bigger than all of the others
put together. We call that the
dominant eigenvalue, and the
corresponding eigenvector we call
the dominant eigenvector, so if you
wait a long time, your vector is going
to look like the dominant eigenvector
times a number that grows like
the dominant eigenvalue to the nth
power. Now, there may be a constant
in front here, and there may be other
terms, but the other terms are smaller
than this term, so if you want to know
what direction are you pointing in the
long run, you're pointing in the
direction of the dominant eigenvector.
And how fast are you growing in the
long run? In the long run, every turn
you're multiplying by the dominant
eigenvalue.
So for example, let's suppose that
we had a matrix, 7 x 7 matrix, and
it had a whole bunch of eigenvalues,
a couple of complex eigenvalues,
some real eigenvalues, and let's
classify which ones are neutral,
and which ones are stable and
which ones are unstable.
Well, -1, the magnitude of -1 is 1,
so that's neutral.
Now, the magnitude of 1 + i
is the square root of 1 squared plus
1 squared, and that's square root
of 2, so this is a complex number
of size root 2, and that's bigger than
one, so that's unstable.
And likewise, the magnitude of 1 - i
is root 2, so that's also unstable.
If you take a power of 1 plus i, it gets
bigger and bigger and bigger and bigger.
You take a power of 1 minus i, it gets
bigger and bigger and bigger and bigger.
On the other hand, if you take a power of
one half, it gets smaller and smaller, so
that's stable. 2 is unstable. -3 is
unstable, because even though it's
negative, negative 3 squared is 9,
negative 3 cubed is -27, negative 3
to the 4th is 81, these numbers are
getting big. They're sometimes positive,
they're sometimes negative, but
they're big. And finally, 0 is as stable
as you could get. After 1 term, it
disappears. 0 to the first power is 0.
So our stable eigenvalues were 1/2
and 0, our unstable eigenvalues were
1 plus or minus i, 2, and -3, and our
neutral eigenvalue was -1.
So which is the dominant one, which
one grows the fastest, well the size
of -3 is 3, the size of 2 is 2, the size
of 1 plus or minus i is root 2.
The biggest of these is -3, so the
dominant eigenvalue is -3.
So the dominant eigenvector is
whatever the eigenvector with
eigenvalue -3 is, and we call that-
that was lambda 6 so this is gonna
be b_6, so if you wait a long time,
x of n is gonna point in the b_6
direction, or maybe minus the b_6
direction, and it gets multiplied
by -3 every term. This term in
the expansion goes exactly like
-3 to the n, times b_6, and that's
the biggest term, so in the long
run, you can ignore everything else.
So here's a picture of stability
in the complex plane. You have the
unit circle, the unit circle is all the
complex numbers of size 1. Those
are the possible eigenvalues that
are neutral. The eigenvalues inside
the unit circle, those have size
less than 1, and so they're stable.
The ones outside the unit circle
have size bigger than 1, they're
unstable. Sometimes they're negative,
sometimes they're positive, sometimes
they're imaginary, but they're all
outside the unit circle is unstable,
inside is stable. Finally, we can talk
about whether a system is stable.
Since in the long run the system
behaves like it's dominant mode,
we call the system stable if the
dominant mode is stable, which
is to say that all eigenvalues are
stable. If you have 17 stable eigenvalues
and 1 unstable eigenvalue,
well the dominant mode is gonna
be the unstable one, so if all
the eigenvalues are small, and by
small I mean less than 1 in size,
the system is stable. If the biggest
eigenvalue has size 1, it's
neutral, or borderline. If the
biggest eigenvalue has size
bigger than 1, it's unstable,
and that's it.
