>> Instructor: Once you've
found the fixed points,
the natural next question
is how stable they are.
To understand this, we need to go back
to our favorite linear dynamical system,
X dot is equal to A times X.
And for the purpose of this discussion,
we're going to just say
that X is two dimensional.
So there's only two
dimensions in the state.
And A is thus a two by two square matrix.
The reason for that will
be obvious in a little bit.
Right now, understand that
it doesn't have to be,
but it helps with visualization
if we're only looking at two dimensions.
Let's say that you have
gone through the trouble
of finding the fixed points
by setting this equation equal to zero,
and you've come up with
your fixed point, XFP.
And what that means is that
when you take the state
and you set it to XFP, you
pop it to this equation,
you're going to get zero.
Alright the state isn't
going to be updated
from that location.
But the question to ask
is when you're not at FP,
how does the dynamical system behave?
Does it move you towards FP?
Does it move you away from FP?
What happens at all points
not at the fixed point?
The question of stability,
fixed point stability,
helps answer that.
Let's say you're moving
infinitesimally small space away
from the fixed point.
Will you go back to it,
or will you run off into
some infinite space,
because the fixed point isn't stable?
You may have heard of a
term called an attractor.
An attractor is a fixed
point that will draw,
draw states outside of
that location back to it.
Now it probably doesn't surprise
you then that the behavior
of this fixed point is
determined by this A matrix
and properties of that A matrix.
Specifically, it's related
to the determinant of A,
and we know how to generally
calculate that, right?
That's the diagonal
and the 2 x 2
and the main diagonal
minus the off diagonal
But, it also is the
product of the eigenvalues.
Similarly it is also related to the
trace of A.
The behavior of these exponents
are really just the trace of A.
And that is actually
the sum of the eigenvalues.
And, so its probably
no surprise to you that if
the behavior of the fixed points
and the stability of the fixed points
is the property of A, and
the key properties of A
are the trace
and those are directly
related to the eigenvalues
Identities the eigenvalues
have a lot to say about the
behavior of the fixed points.
And if you came to that conclusion,
you'd be correct.
Now we can take a look at this
in much more concrete terms
looked specifically at how the properties
like determinate, trace and eigenvalues
determinate control the
fixed point behavior.
This is a poincare diagram.
It will tell us,
at least for two dimensions,
the full behavior of fixed points.
And so you can see that there's
five major sections to this mapping.
Each of them corresponds to a different
kind of fix point.
There's also fix point
behaviors that are along these
lines as well
that help us understand what's going on.
So, for example
the trace of A
is zero
and the determinate is positive.
Then we are in a situation
where you have a center stable
fix point.
What that means is
if you are not on the fix point
but somewhere off
what you will do is you will rotate
and oscillate around the fix point
as you evolve over time.
Similarly, if you are over here
where you have the trace of A negative
and a positive determinate
you are, you are, uh
above this discriminate line here.
Then you're going to end
up with a spiral sink.
A fix point that was stable
and if you are off of it you will
slowly spiral around and around
until you collapse into the fix point.
Similarly if you are on this
side of the discriminate
line, right.
Where the trace is negative
and the incriminate is positive
you will end up with a sink
where you will just eventually fall
into the point if you are not on it.
Now these are written in
properties of the determinate
and the trace.
You can also
look at this map
with the same sections as the
function of the eigenvalues
because again, we mention
that the eigenvalues
are related to properties
like the determinate
and the trace.
And since we're only talking about
a two-dimensional system
this gives you all the
potential options you can have
in two dimensions.
And, there is only a
maximum of two eigenvalues
that you can have in two dimensions
because a full rank two-dimensional matrix
has only up to two eigenvalues.
Now,
if the eigenvalues
are real
then you are below this line.
This discriminate line.
What that means, that discriminate
is that, is that
is that portion under
the radical
of the characteristic polynomial.
And what this is saying
is that anywhere below this line
below this curve of the discriminate
the eigenvalues are real.
Here, eigenvalues real.
And, here, the eigenvalues are complex.
So if you know the eigenvalues
are complex, you're only in this space.
If your eigenvalues are real
then they are somewhere down here.
Okay, so that helps you to determine
whether or not
your in
a spiraling situation or
a non-spiraling situation.
If your eigenvalues are complex
your spiraling or
oscillating in some fashion.
If your eigenvalues are real
you're simply falling in
an exponential fashion.
Either falling in, or pushing a way out.
Now if your eigenvalues
are real
but they're both negative
then you are in this regime
of the map.
And, you're behavior of the fix point
is the sink.
That's called a stable
node or an attractor.
On this end over here
if your eigenvalues
are both greater then zero
and your eigenvalues are real
because again, were below this line
then you have a source.
One in which anywhere
outside the fix point
you are going to explode off
into infinity.
If you think about what eigenvalues
mean with the respect
to the characteristics
polynomial describing E to the
E to the
E to the eigenvalue it
actually makes sense.
Right?
E to the positive number
is going to be big.
E to the negative number
is going to shrink and decay into zero
and that is your behavior sinks into these
sinks or sources.
Now, you fall into this regime down here
if your eigenvalues
are of opposite signs.
If one is negative and
the other is positive
then you are below this line over here
and you have the behavior
of a saddle point.
That leave us with just
these regions over here to define.
And you can probably guess what these are.
Because
we know that are
eigenvalues are complex here
which means that they're complex
contingents of each other
that means the imaginary
components have to be
opposite signs
and so the only thing left
to do and think about then
is real components.
And if you think about
how the real component
was set up here versus here
then you have the same situation.
If your eigenvalues are
negative and complex
then you have
this mapping here
and you find yourself
at
a spiral sink.
Where as if your positive
component of your eigenvalues
sorry, if they're positive
then your going to find
yourself with a spiral source.
They'll be in this region over here.
That's it.
They're the only identities
that you can have.
You've either got positive
or negative eigenvalues
and they are either
real or complex.
And, if they have opposite signs
you'll find yourself at a saddle point.
Note that, you can't find
yourself in a situation
where you have opposite
signed real eigenvalues.
That opposite signed eigenvalues
'cus those are complex conjugates.
So, there's no negative dip on this side
to plot.
But this is it.
This then will give you the behavior
of your dynamical system
as it relates to the
stability of your fixed point
or fixed points.
