So in this question, we are asked to revisit a system that we had in Question seven, which is ex prime is equal to X minus two y plus five x y and why prime is equal to two X plus y. Okay, um, and were asked to sketch your face plane for this. Ask them. Ah, and remember, purity went through the system, found that the Jacoby in is gonna be, um, one plus five. Why? Negative, too was fun x two and one and that we have some equilibrium. The first is 00 The origin with Eigen values one plus three minus two. I, um and that means that this is gonna be an unstable spiral. And then the other fixed point here is, uh, NATO's Excuse me, 1/2 negative one. And that has William to equal to negative three halves, plus or minus the square root of 29/2. And that means that we have a saddle here. One of those is positive. One of this is negative. And of course, they're both riel. So we have a saddle eso weaken. There's a couple ways that we can approach making this fix the space plane. Ultimately, we're going to use technology. We use a face plain illustrator to find the full picture. But we can. There are some things we can do without technology that will help us, um, get an idea what the baseball is gonna look like. And the one that I wanted to talk about is using our know Klein's toe illustrate what this is gonna look like. So, um, are no clients are going to win Ex Prime equals zero. And why practical zero. So an ex prime equals zero. That means that X minus two. Why was five x y equals zero and, ah, when why Prime equals zero. That means that excuse me, that you read the scale that means two x plus y equals zero. Sorry. Got lost in my notes. And so we can these these air equations for lines in the X y plane. Um and so anywhere along this line are, uh, you know, the delta X Delta y vector is going toe have no change in X right. It's only going to be a change in why? So it's gonna be a straight up or down, and on this no climb, it will be only, uh, left, right, Right. I'm so you can kind of piece together with this image is gonna look like basing these no plan. So the easiest way to do this would be toe to solve both of these for why? So, um are why no client is pretty easy. We have, uh, two x plus y equals zero. That means that why equals negative two X, right? This is just a nice line. That is, they were pretty used to um uh, the X No client, on the other hand, is a little bit more difficult. We have X minus two y plus five x y equals zero to sell for. Why? What we're gonna do is, um, see that X is equal to to my minus five x y benefactor out of why to yet tu minus five X. Uh and then I'll say, excuse me, getting in myself. And now we can see that why is equal to X over two minus five X. Um, cool. So now we have two lines that we can plot and use that Teoh get an image of our face plane. So one of these lines y equals negative two x is quite simple. You know, all underlying this and written dry it in red. Um, it's just this line. Right. Um, this other one y equals x, uh, minus X over two minus five X eyes going to look a little bit different, but we we do know that this has a nascent toot at 2/5 right? So has a nascent it here. And we see that before 2/5. Right when exes is less than 2/5. Um, well, I guess easier. Easier to see when X is greater than 2/5. This is going to be negative. So I'm gonna draw this in green. So it's going to look like this. Buy something correctly. And, um, before that, before defense, it'll look something like this. Right? So you have, uh, to points were these curves intersect, and those are are fixed points, right? And that this assures us that are drawing is a little bit is at least a little bit reasonable, because we know that we have a fixed 0.0 and a fixed point at this point, which we might call 1/2 negative one that that looks reasonable. So that's good. Now we can use these no clients to kind of fill in what we expect. Uh, what we expect this to look like. So remember, on our red No, Klein, that is where the changing why or why prime is equal to zero. So any vector that is on the final climb is going to be, um, just straight up and down or especially straight side to side. It's only going to have change in the X direction. Um, and so we can think about a point here, right? In this case, ex, um, we think about the change in X here. If we have a negative accident, a positive. Why we have this term is negative. This term is negative, and this term is negative. So these are all going to be negative. So any vector it starts a long the y know climb is going to only move in the extraction on that means, uh, well, that means that, um and because we know that when we play the point in from this quadrant into our our equation for ex prime, it's negative. We know that, uh, it's going to be in the negative X direction. So we can continue to do that for other parts of our y know. Client, for instance, Here we can What points in between Where X is between, um, zero and 1/2 and wise between zero native one. And we see that this is positive, right? So x x is positive. Um, why is, uh, negative one? So why this term here? Negative two times something that's between zero native one is also positive. Um, and we know that it's going to be these two. Some together is gonna be greater than five x. Why? So we know that this is going to be positive. And then again, when you cross over that line, we know that these we're going to be negative like that. That's how I kind of feel in this caress, um, now, on our external climbing over vectors air going to either pointing up or pointing down right, because the change in X zero so we can feel that in. And this one's gonna be a little bit easier, because what we're comparing it to is just the the equation for why prime, which is just a nice linear equation. So, um, if why is greater than negative. Two X right. So that that would be above our red line here. So if y is greater than negative two X, Um, that means that this is going to be positive, Right? So in this, when y is greater than negative two x are vectors on the X. No, Klein went straight up. That and then in the opposite case, they're gonna want straight down. I think this, uh and in this way, we have kind of filled in What are baseline might look like, right. We can fill in the areas in between by, you know, just composing these vectors, right? We have a negative x negative. Why in this region, eso the vectors here are gonna be pointing in that general direction in this region, we have positive Y negative X. So are vectors are gonna look like this in this region? We have negative. Why positive acts that we're gonna look be looking like this. And in this region, we'll be looking like that. Eso we kind of fill in What? The citizen. Now we can kind of see this unstable spiral At 00 we can have a new idea of which way it's going to spiral out, right? It's going toe spiral out like this. Probably. Um, we also see that the saddle point we can see kind of the stable and unstable directions, right are stable. Direction is gonna probably be coming in this way and are unstable direction. Looks like it'll be going out that way. Uh, so we can use that information to kind of direct our or a graph here, and we can try to plot some. We can plot some solutions a little bit. Uh, that or maybe gonna look like this and like this. Um, yes. Oh. So once we have this, we can, uh, get an idea of what our face Porter's gonna look like and how this is really crowded because this is a drawing and not a nice computer output. So let's switch over to now or computer output. And this this is going to look like this. So I'm using P plane eight, which is a camp creator made by some professors at Rice University. Um, which is used was originally published in Matt. Lab is now available of Java. Um, and we use it to fine face planes for systems international equations so I'm going to have it Graphic face creams you already saw. But we see now what are know Klein's look like, right? Are the red No, Klein Is the this why? Excuse me? The external clients. So any at any point on the X no climb this red line is going Teoh be associated with a better. That's playing straight up straight down. And then this yellow line was just harder to see is the, um why no clients or any point on this wine offline is gonna be associated with the vector pointing straight left or straight. Right. But we can fill in some solutions here, and we see that are sketches pretty good. We have this spiral moving in the direction that we expected to. And also we have a, um the saddle point, which has the stable directions that we expected it to, and the unstable directions which we expected at the help eso We can fill our face playing up with solutions. Teoh get a nice idea of how things flow. And I'm I'm just doing forward solutions. But this calculator can do forward and backward solutions, um, to see stable and unstable attractors here So, in any event, um, this is a nice calculator. Teoh use, uh, have visualized base points like this. But if you don't have access to something like this, we can still get a pretty good sketch from a face plane just by, um, using the no clients. And there's other ways we can. We could have used the Eigen vectors for this saddle point to find the stable and unstable directions. Um, and that would have informed a little bit about and from us a little bit. About what, uh, how the spirals going to spin as well. So there's a lot of ways that we can sketch this face plane, but of course, using technology like this is is the fastest and easiest.