So let's look at the last
type of phase portrait. One for spiral points, here the differential
equation is x_1 dot equals minus
one-half x_1 plus x_2, x_2 dot equals minus
x_1 minus one-half x_2. The matrix is minus one-half
1 minus 1 minus one-half. If you do the eigenvalue and eigenvector analysis
on that matrix, you get a complex
conjugate eigenvalues. So one of the eigenvalues
is minus one-half plus i and its associated
eigenvector is one i, and then you have the
complex conjugate pair. If we write the solution by constructing two real solutions, we'll end up with x equals this decaying exponential times a times this cosine
t minus sine t plus b times the sine t
cosine t. This type of solution is circular and then it's having a
exponential decay, so this is why you get a spiral. So let me show you what the
solution might look like. So the origin is the fixed point and its
decaying exponentially, so the fixed point is stable and the solution spiraling
into the origin. So it could look
something like this, and then you spiral
into the origin, and because it's
stable it's going in. But we're not really sure
this is what it looks like because there's two ways
to spiral into the origin. This is one of the ways, here you see that the motion
is if you look like a clock, this is clockwise, so this
is a clockwise spiral. On the other hand, if we have a counter
clockwise spiral, so remember this is x_1
and x_2, x_1 and x_2, a counterclockwise spiral will be spiraling into the origin, but in the other
direction, like so. So the solution can
look like one of these, it's a spiral into the origin, but could be clockwise or in this case, it's
counterclockwise. So how do you tell
which one it is? The simplest way is just to look at the point on the x_2 axis. So here we have x_1 equal to 0, at that point, and here also. If we look at x_1 equal to 0, then we have x_1 dot equals x_2. So x_1 dot equals x_2 meaning that you're
moving to the right. So in the top diagram, the trajectory is moving
towards the right, and the bottom diagram, the trajectory is
moving to the left. So we substitute in x_1
equals 0, x_2 is positive, then x_1 dot is positive here, and that should be,
that's correct. But here x_1 dot is negative, so this one is incorrect, this one is correct. Okay. Let's look at a computer-generated phase
portrait using MATLAB. So here you see how
the trajectories are all spiraling into the origin. Okay. So let me review, in this last and final case, we considered spiral points. Here the eigenvalues are complex, they show up as complex
conjugate pairs. The key here is the real
part of the eigenvalue. So if the real part is negative, then this is a stable spiral, all solutions spiral
into the origin, if the real part is positive, then this is an unstable spiral, all solutions spiral
out of the origin. Then you have two
choices whether it's a clockwise spiral or a
counterclockwise spiral, you can determine
which one it is by examining the
differential equation. I'm Jeff Jasanoff,
thanks for watching. I'll see you in the next video.