AIHL 5.17.4 Sketching phase portraits introduction | Free Mathematics Applications & Interpretation (AI) Video | RevisionDojo
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AIHL 5.17.4 Sketching phase portraits introduction Learn AIHL 5.17.4 Sketching phase portraits introduction in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 5.17—Phase portrait.
About this video Learn AIHL 5.17.4 Sketching phase portraits introduction in this free IB Mathematics Applications & Interpretation (AI) video lesson for AHL 5.17—Phase portrait.
The video discusses different types of phase portraits for coupled linear differential equations, focusing on the role of eigenvalues in determining the stability and behavior of solutions.
Key types of phase portraits include:
Saddle Point : One positive and one negative eigenvalue, leading to unstable behavior.Source : Two positive eigenvalues, resulting in solutions diverging from the origin.Sink : Two negative eigenvalues, where solutions approach the origin, indicating stability.Spiral Source : Complex eigenvalues with a positive real part, causing solutions to spiral outward.Spiral Sink : Complex eigenvalues with a negative real part, leading to solutions spiraling inward and approaching the origin.
The video emphasizes the importance of understanding these different cases to effectively sketch phase portraits and analyze the stability of systems.
Video transcript 00:00 Hi guys, okay, so in
00:02 the last video we solved
00:03 coupled linear differential equations and
00:08 we sketched the phase portrait
00:11 of the solution and in
00:12
this case it was a
00:14 solution where the origin was
00:18 this this was actually the
00:19 sketch of the phase portrait
00:21 that we did. However, there
00:24 are different types of phase
00:27 portraits that we need
00:29 sketch. So this is one
00:30 of them, the saddle. And
00:33 these are the five I'm
00:34 going to do separate videos
00:35 for each one of them,
00:38 kind of just one video
00:39 where you saw them all
00:40 together and you can see
00:42 perhaps why they end up
00:47 like this. And then when
00:48 you watch the video for
00:50 each of them, you'll get
00:51 a better understanding of what's
00:52 going on. So basically the
00:56 cases are for the different
00:59 eigenvalues. I've put the formula
01:01 for the solution of the
01:04 couple differentially with equations up
01:06 here. Now if you remember
01:08 the lambda 1 and the
01:10 read it, where the eigenvalues.
01:13 So the eigenvalues play a
01:15 very important point, a very
01:17 important part in how these
01:20 face portraits look. So I'll
01:23 go straight to the saddle
01:24 And the saddle point where
01:26 the origin was a saddle
01:28 point, the eigenvalues were real
01:32 and one was positive and
01:34 one was negative. And you
01:35 end up getting something that
01:36 looks a bit like this.
01:37 The blue lines were the
01:39 eigenvectors. If, however, you have
01:44 two eigenvalues and they're both
01:45 positive, what happens is you
01:48 get the same two blue
01:49 lines with your eigenvectors, but
01:51 it doesn't look like
01:52 this, it looks like this
01:54 because it comes the solutions
01:55 come out of the origin
01:56 and they kind of follow
01:57 one of the eigenvectors. Well
02:01 they originally followed this eigenvector
02:04 off and follow this eigenvector.
02:05 The bigger eigenvalue will determine
02:11 which eigenvector they follow. And
02:15 into a lot more detail
02:16 of that when I actually
02:20 source and it's unstable. So
02:22 two positive eigenvalues, you get
02:24 a source unstable and it
02:26 looks something like this. If
02:28 you have two real negative
02:30 eigenvalues, they're both negative, you
02:32 get a sink, it looks
02:34 something like this and we
02:35 say it's stable, it's stable
02:37 because the solutions are all
02:39 approaching the origin. So if
02:41 it gets knocked off the
02:42 origin, it'll just go back
02:45 into the origin. Here, if
02:46 it gets knocked off the
02:48 fly away. Okay, this is
02:54 the saddle. One is positive,
02:55 one is negative and this
02:56 happens. If the eigenvalues are
03:00 complex and the real part
03:03 of the complex eigenvalues are
03:08 positive, so the real part
03:09 is positive. What you get
03:11 is the spiral source and
03:14 we say it's unstable, so
03:16 is it starts the solution
03:19 spiral out like this. The
03:23 reason that happens again, I'm
03:25 going to more detail, but
03:25 because the real part is
03:27 both positive, this e to
03:32 a positive number t just
03:34 causes the solutions to move
03:41 away from the origin or
03:44 here if we have complex
03:48 eigenvalues but the real part
03:50 is negative a bit like
03:52 this they spiral in so
03:54 it's the complex part that
03:55 causes it to kind of
03:56 rotate and the negative part
03:59 of the real the negative
04:02 real part of the complex
04:04 number causes it to approach
04:06 zero because it's actually the
04:09 solutions are decaying if you
04:11 like an approaching zero
04:12 we say it's stable 0,
04:14 0 is a stable point.
04:16 And finally, if it's a
04:18 complex number and there's no
04:19 real part or the real
04:21 and the solution is just
04:25 the trajectories. So these are
04:28 all trajectories, by the way.
04:29 They just go around the
04:33 origin, but they don't actually
04:34 spiral into the origin. So
04:37 that's basically it, guys. The
04:39 different eigenvalues, real plus plus
04:40 real negative, real plus negative
04:42 and then the complex, but
04:44 it's just the real part
04:45 is positive, negative and zero
04:47 and you end up getting
04:48 this type. Well, this is
04:54 what the origin is. That
04:55 point, it's a source, sync,
04:57 subtle, spiral source, spiral sync,
04:59 and center. We say it's
05:00 an unstable point, a stable
05:02 point, an unstable point, an
05:03 unstable point, an unstable point,
05:04 and a stable point, and
05:05 the graphs, the sketches of
05:08 which are these kind of,
05:10 well the red here in
05:11 the purple here and the
05:13 blue here these are trajectories
05:14 and the actual sketch is
05:16 called the face portrait and
05:18 say guys, I'll do, well
05:20 I've already done this one
05:20 but I'll do one, two,
05:21 three, four, five more videos
05:23 lot more detail about how
05:27 to draw these face portraits.
05:29 Okay, that's it guys, I'll