All right, so we draw face portrayed for the matrix of Given. First, we gotta find the equilibrium points, and we do that by first converting into the respective equations. So it's two x plus three. Why? And minus X minus two. Why? So you are finding the equilibrium point? Um, a trivial equilibrium point for this one is actually zero comma zero.
And you can see that because it's both X and y are zero. Then both equations are equal to zero at the same time, which is the condition of being an equilibrium point. Um, I don't think any other point works. So equilibrium point is zero comma zero no less star sketching with face worker. Oops.
I was terribly not straight line. Who? Okay, I'm so sorry. We try something else. I figured I could draw half a line. Street shouldn't be as hard.
All right. This is probably the best incredible to get a while. All right, so we know that this is the equilibrium point where we're gonna be focusing at now and find Eigen values. So for that, we go back. The matrix that is aimed is equal to 23 naked of one and negative, too, to find the Eigen values.
Behalf. It's a practice with the identity matrix Times lambda, which is Landau 00 lambda. And we get this important matrix, which is tu minus lambda three. Yeah, minus one and minus two minus Lambeau. Because this is important.
I'm gonna circle this, they have to come back to it. And then we have to find the characteristic equation, which is the diagonals more multiplying each other. It's a practice. So this case, it's to minus Landau times minus two minus lambda minus three times minus one. This becomes two times negative.
Two is negative. 42 times minus. Lambda is minus two Lambda Negative Lambda. Minus times plus two people two plus two. Lambda Native, Longer time thinking love is gonna plus Lambda Square minus native.
Three sets of plus And we just said this equal to zero These bold cancel out This becomes negative one and this state's latest. So we have Lambda squared minus one equals zero, which implies that land, uh, this equal to plus or minus one. These are two Eigen values. Lambert is equal to one Onda Lambda is equal to negative. One probably didn't give too much space for this one right here.
We can part to shoot them later. All right, So once you have your own values, how do you find Agon vectors? It's not too hard, actually, but it does take a little bit of work. You go back to our special matrix right here and substitute the Eigen values a tu minus one. This one three and negative one on minus two, minus one is minus three. And we have subtracted with all right.
Not subtracted. Multiplied. If they're Eigen Vector and you have to get 00 for that to be a writer. Victor. Yeah, We've finally been vector figuring out what number does that for us.
So we have, well, playing them out. We get a plus three B p is equal to zero. Negative. A minus three B is equal to zero. So if you look at this, you could see that, um negative three and one should work.
A is equal to negative three. And be physical. 21 If you substitute these values, you can see that that should give you zero. So our first I in Vector was right in red is negative. Three and one are sinking.
Hiding. Victor, You know how much space Weird fire who does not that bad. All right. And we do the same thing, except we have to now substitute with negative one. So two minus negative.
One would be three three, a negative one, Onda. Negative two minus negative one, which is negative. Two plus warm, which is native one. Same procedure. Um, we get three A plus three b is equal to zero and negative a minus.
Negative. Right are just negative. Eight night minus B is equal to zero. Um, for this one, let's see if we have a is negative one. This would be one.
And if we have B is one B minus ones, that b zero. All right, So for this one, the idea of value dying vector is going to be negative. One and one. All right, so we found our Ryan makers. Cool.
Aren't too shabby. Not too shabby. All right, so we know our equilibrium point, and we know our wagon. Beckers. Um, So you have negative three and one for one.
All right. So negative three in this direction. Some making these points show extraction going. It's a negative three on day one, so I'm going in this direction. All right, so that was just show you what? Directions.
I'm gonna race that because it's gonna be really messy. All right, um, that is our for a standing. Becker. Um, this Eigen value was positive, so that means it's going to be in this direction. All right.
Um, the next one was negative. One and one and clearly to show you what direction I'm going in. Negative one on one. Um, it looks like this is going the same direction. Except it's not as aggressively in the X direction.
So I guess it would be a more steeper line. All right, we can. Rather before I do that, let just quickly note this is our again value one. This line was for one, all right. And this one is three times more steep.
Then the line, we just true it was my wow. That's so bad. All right, that's the best I could do. And this was or I could value negative one. Since this negative, this will actually be going in this direction.
Two words, the origin. All right. That tells us a lot of information. Actually, So, since this is what we call a saddle, we don't have to be too fussy about the strength of the currents. We just have to know that are struggling in red.
If I drop a stone here, this will dominate it. Until I get to this, I Oh, God. Uh, this will dominated until I get to this Eigen vectors region which will start dominating. And over there, if I more over here, I will be going in this direction until I reached the second value. Just push me in that direction.
Hopes me going more this way and that I turned that way. I'm over here rather if I'm over here, uh, we pushed here until that I get in, Victor gets stronger. I go in that direction. Yeah. This is not going to be the most prettiest various portrayed.
But I hope you get the point. I'm Iraq and you drop me to the river over here when I first load that way and then go this way because then it gets too strong for me to fight that current. Yeah, so I didn't mean these currents are not supposed to touch each other at all. I'm just a botanic artists, which is why they touch each other. But this is what a general phase border would look like.
Um, you could use some software, like wolf Mouth or something to get a Niedere portrait. But the basic point for this portrait is that this is a saddle point. Onda, Uh, this is what would roughly look like if you if you have the directions off your again vectors hammered out properly, then doesn't really matter because, um, you could figure out directions to draw them. So, yeah, that's the important thing. All right.
Thank you..
