AIHL 5.17.4 Sketching phase portraits introduction

00:00Hi guys, okay, so in

00:02the last video we solved

00:03coupled linear differential equations and

00:08we sketched the phase portrait

00:11of the solution and in

00:12this case it was a

00:14solution where the origin was

00:16a saddle and it was

00:18this this was actually the

00:19sketch of the phase portrait

00:21that we did. However, there

00:24are different types of phase

00:27portraits that we need

00:28need to know how to

00:29sketch. So this is one

00:30of them, the saddle. And

00:33these are the five I'm

00:34going to do separate videos

00:35for each one of them,

00:37but I wanted to do

00:38kind of just one video

00:39where you saw them all

00:40together and you can see

00:42perhaps why they end up

00:47like this. And then when

00:48you watch the video for

00:50each of them, you'll get

00:51a better understanding of what's

00:52going on. So basically the

00:55difference

00:56cases are for the different

00:59eigenvalues. I've put the formula

01:01for the solution of the

01:04couple differentially with equations up

01:06here. Now if you remember

01:08the lambda 1 and the

01:09lambda 2 if you can

01:10read it, where the eigenvalues.

01:13So the eigenvalues play a

01:15very important point, a very

01:17important part in how these

01:20face portraits look. So I'll

01:23go straight to the saddle

01:24one.

01:24And the saddle point where

01:26the origin was a saddle

01:28point, the eigenvalues were real

01:32and one was positive and

01:34one was negative. And you

01:35end up getting something that

01:36looks a bit like this.

01:37The blue lines were the

01:39eigenvectors. If, however, you have

01:44two eigenvalues and they're both

01:45positive, what happens is you

01:48get the same two blue

01:49lines with your eigenvectors, but

01:51it doesn't look like

01:52this, it looks like this

01:54because it comes the solutions

01:55come out of the origin

01:56and they kind of follow

01:57one of the eigenvectors. Well

02:01they originally followed this eigenvector

02:03and then kind of go

02:04off and follow this eigenvector.

02:05The bigger eigenvalue will determine

02:11which eigenvector they follow. And

02:14as I said, I go

02:15into a lot more detail

02:16of that when I actually

02:18do a video of this.

02:19This is called a

02:20source and it's unstable. So

02:22two positive eigenvalues, you get

02:24a source unstable and it

02:26looks something like this. If

02:28you have two real negative

02:30eigenvalues, they're both negative, you

02:32get a sink, it looks

02:34something like this and we

02:35say it's stable, it's stable

02:37because the solutions are all

02:39approaching the origin. So if

02:41it gets knocked off the

02:42origin, it'll just go back

02:45into the origin. Here, if

02:46it gets knocked off the

02:47origin, it

02:48fly away. Okay, this is

02:54the saddle. One is positive,

02:55one is negative and this

02:56happens. If the eigenvalues are

03:00complex and the real part

03:03of the complex eigenvalues are

03:08positive, so the real part

03:09is positive. What you get

03:11is the spiral source and

03:14we say it's unstable, so

03:15the spiral source

03:16is it starts the solution

03:17start at 0, 0 and

03:19spiral out like this. The

03:23reason that happens again, I'm

03:25going to more detail, but

03:25because the real part is

03:27both positive, this e to

03:32a positive number t just

03:34causes the solutions to move

03:41away from the origin or

03:44divergent.

03:44here if we have complex

03:48eigenvalues but the real part

03:50is negative a bit like

03:52this they spiral in so

03:54it's the complex part that

03:55causes it to kind of

03:56rotate and the negative part

03:59of the real the negative

04:02real part of the complex

04:04number causes it to approach

04:06zero because it's actually the

04:09solutions are decaying if you

04:11like an approaching zero

04:12we say it's stable 0,

04:140 is a stable point.

04:16And finally, if it's a

04:18complex number and there's no

04:19real part or the real

04:19part is 0, why you

04:20get it as a center

04:21and the solution is just

04:25the trajectories. So these are

04:28all trajectories, by the way.

04:29They just go around the

04:33origin, but they don't actually

04:34spiral into the origin. So

04:37that's basically it, guys. The

04:39different eigenvalues, real plus plus

04:40real negative, real plus negative

04:42and then the complex, but

04:44it's just the real part

04:45is positive, negative and zero

04:47and you end up getting

04:48this type. Well, this is

04:54what the origin is. That

04:55point, it's a source, sync,

04:57subtle, spiral source, spiral sync,

04:59and center. We say it's

05:00an unstable point, a stable

05:02point, an unstable point, an

05:03unstable point, an unstable point,

05:04and a stable point, and

05:05the graphs, the sketches of

05:08the children

05:08which are these kind of,

05:10well the red here in

05:11the purple here and the

05:13blue here these are trajectories

05:14and the actual sketch is

05:16called the face portrait and

05:17that's it. So as I

05:18say guys, I'll do, well

05:20I've already done this one

05:20but I'll do one, two,

05:21three, four, five more videos

05:23where I go into a

05:23lot more detail about how

05:27to draw these face portraits.

05:29Okay, that's it guys, I'll

05:30see you next video.