Hey guys, in this video I'm going to go
over how to draw the trace-
determinant plane and how to use this plane to figure out how to classify a phase
plane portrait. To keep the video short, I'm not going to go over
stability of these points. I'm just going to go
over how to classify the type. Whether it's center, spiral, saddle point, and etc. And I'll go over stability
maybe in another video. This is a nice graphic. I actually used this when I took
Differential Equations. I definitely didn't create it myself. I got it off the internet and I just cut and pasted it from some
powerpoint slides. I will put the link to those slides down in the description box below.
And this is the trace-determinant plane that I liked to study when I was memorizing how to
determine the phase plane portraits. Okay so how
do you draw a trace-determinant plane and what are the important parts that you have to remember? It's basically an x and y-axis
where the x is instead the trace and the y-axis is the determinant.
Any determinant that is negative is going to be a saddle point. You don't have to continue
the axis to the third and fourth
quadrants. You can just continue it up from the origin. And then you have a parabola, which is D is equal to
trace squared over four.
And you need to know this formula because you're going to have some cases where you have to determine
if it's in the spiral zone or
if it's in the nodal zones. And we already said that any determinant is a saddle point and then
we have that the nodals are below the parabola.
Anytime a trace is positive you're going to have a source. Anytime the trace is negative you're going to have a
sink, and then what're inside the parabola are spirals. And then you
also have border cases. If something is on this line, which is the determinant
axis, then it's going to be called a center.
If something falls on this
curve--if it's positive, so the trace is
greater than zero, it's going to be considered a nodal source.
For my class all we needed to call it was a nodal source but this graph
specifies that it's considered a degenerate nodal source. If the trace is
negative, then it's going to fall under the nodal sink and again this graphic specifies that it's considered a
degenerate nodal sink. If something falls on this trace axis, the
x-axis, it's going to fall as a saddle point or this graphic is calling it a saddle node that's either stable or unstable.
For my class,
we were okay with just calling it a saddle. And so if you can remember how to draw this and the
labels that I have here, then you should be able to determine the phase plane
portrait for any type of linear system that they give you. I'm just going to go over a few examples now.
The first example I have is the coefficient matrix [8 5;
-10 -7].
What you always have to do for these problems is
calculate the trace and the determinant and then figure out where these points would lie on the trace-determinant plane.
The trace is going to be 8 + 7, which is 1. The determinant is 8 * -7 minus
-10 * 5 and that's going to be equal to -6. If we plot the point (1, -6),
we're going to be on the right-hand side of
the origin for the x-axis and on the
negative or bottom side of the y-axis. That's going to be a saddle point.
The next coefficient matrix I have is [-2 0;
1 -1]. The trace for that is going to be -3.
The determinant for that is going to be -2 * -1 minus 0. That's 2.
We have the point (-3, 2).
-3 will take us to the second quadrant and then 2. This is one of the problems where
we have to calculate what is the parabola to determine whether it's a nodal or spiral sink. T squared divided by 4
is equal to -3 squared
divided by 4, which is 9 divided by 4, which is greater than two. This is greater than two,
which means that we are in the nodal sink area.
For problem number three,
we have the matrix [-10 -25;
5 10]. For that, the trace is going to be 0 and the determinant will be  -10 * 10 minus
5 * -25, which is equal to
25. We have the point (0,
25). We go to 0 on the x-axis or the trace and we go up. That's going to be a center.
This is the last problem I have. The trace for this is -4 and the determinant is
2 * -6 minus -4 * 8, which is 20. We have the point
(-4, 20). That's going to be on the left-hand side again. And again
we have to determine T squared divided by 4. That's going to be -4 squared divided by 4. This is
16 divided by 4, which is equal to 4. This value will be 4 when T is
-4. Let's say that's over here. Twenty is way above 4.
We are in the spiral sink zone. This is a spiral sink.
That's the last problem that I have for these types of questions. Hopefully you see that by using
this graph, it makes it really easy to remember how to
classify the phase portraits.
*chiptune music*
