[{"Name":"Phase Plane","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Phase portrait of Linear Systems","Duration":"3m 22s","ChapterTopicVideoID":10355,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/10355.jpeg","UploadDate":"2017-10-17T05:15:30.0300000","DurationForVideoObject":"PT3M22S","Description":null,"MetaTitle":"Phase portrait of Linear Systems: Video + Workbook | Proprep","MetaDescription":"Phase Planes - Phase Plane. Watch the video made by an expert in the field. Download the workbook and maximize your learning.","Canonical":"https://www.proprep.uk/general-modules/all/ordinary-differential-equations/phase-planes/phase-plane/vid10705","VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.865","Text":"In this clip we\u0027re starting a new topic of the phase plane and"},{"Start":"00:05.865 ","End":"00:12.105","Text":"something called phase portraits of linear systems of differential equations."},{"Start":"00:12.105 ","End":"00:17.280","Text":"It turns out that certain types of systems have a nice way of representing them,"},{"Start":"00:17.280 ","End":"00:21.760","Text":"graphically, drawing them somehow in the plane."},{"Start":"00:21.760 ","End":"00:26.680","Text":"The type of systems we\u0027re going to consider are;"},{"Start":"00:26.680 ","End":"00:29.160","Text":"linear, homogeneous,"},{"Start":"00:29.160 ","End":"00:33.250","Text":"planar, with constant coefficients."},{"Start":"00:33.940 ","End":"00:38.010","Text":"I\u0027ll show you what a general one looks like."},{"Start":"00:38.170 ","End":"00:42.500","Text":"Here\u0027s the system we\u0027re going to study."},{"Start":"00:42.500 ","End":"00:46.820","Text":"The bold usually indicates vector or matrix,"},{"Start":"00:46.820 ","End":"00:50.660","Text":"x\u0027 is matrix A times x,"},{"Start":"00:50.660 ","End":"00:56.030","Text":"where A is a 2 by 2 matrix consisting of constants,"},{"Start":"00:56.030 ","End":"01:01.939","Text":"and x is a 2 dimensional column vector,"},{"Start":"01:01.939 ","End":"01:03.950","Text":"usually x_1, x_2."},{"Start":"01:03.950 ","End":"01:07.260","Text":"Sometimes we\u0027ll use x and y."},{"Start":"01:08.030 ","End":"01:11.655","Text":"It\u0027s a vector function of t,"},{"Start":"01:11.655 ","End":"01:14.340","Text":"t is typically from physics so"},{"Start":"01:14.340 ","End":"01:19.085","Text":"this time doesn\u0027t have to be and from minus infinity to infinity."},{"Start":"01:19.085 ","End":"01:21.670","Text":"Now it\u0027s planar because it\u0027s in 2 dimensional,"},{"Start":"01:21.670 ","End":"01:25.310","Text":"it\u0027s linear because the system is linear,"},{"Start":"01:25.310 ","End":"01:30.440","Text":"homogeneous, because there\u0027s no plus some constant."},{"Start":"01:30.440 ","End":"01:31.945","Text":"It\u0027s just Ax."},{"Start":"01:31.945 ","End":"01:35.825","Text":"That\u0027s the system we\u0027re going to try and represent graphically somehow."},{"Start":"01:35.825 ","End":"01:40.940","Text":"We\u0027re going to assume that our matrix A is non-singular."},{"Start":"01:40.940 ","End":"01:45.900","Text":"It\u0027s just not interesting and there are easier ways of solving it if it is singular."},{"Start":"01:46.490 ","End":"01:50.270","Text":"In this example, we take the matrix A to be 1,"},{"Start":"01:50.270 ","End":"01:51.470","Text":"2, 3, 0."},{"Start":"01:51.470 ","End":"01:53.600","Text":"You can check it\u0027s non-singular,"},{"Start":"01:53.600 ","End":"01:55.025","Text":"and if you write it out,"},{"Start":"01:55.025 ","End":"01:57.230","Text":"write x as x_1, x_2,"},{"Start":"01:57.230 ","End":"02:00.500","Text":"we\u0027ll get this system of equations."},{"Start":"02:00.500 ","End":"02:03.585","Text":"But x_1 and x_2 are functions of"},{"Start":"02:03.585 ","End":"02:11.100","Text":"t. We\u0027re leading up to an example of something called the phase portrait."},{"Start":"02:11.100 ","End":"02:14.240","Text":"I\u0027ll say a few words about it first."},{"Start":"02:14.240 ","End":"02:21.560","Text":"Now a phase portrait of the system is just a representative set of solutions."},{"Start":"02:21.560 ","End":"02:24.740","Text":"There\u0027s a lot of discretion in the word representative,"},{"Start":"02:24.740 ","End":"02:26.210","Text":"not too many, not too few,"},{"Start":"02:26.210 ","End":"02:28.580","Text":"there are some that will just give you an idea."},{"Start":"02:28.580 ","End":"02:30.380","Text":"Once you\u0027ve seen the first example,"},{"Start":"02:30.380 ","End":"02:34.310","Text":"you\u0027ll know what I mean by representative set of solutions."},{"Start":"02:34.310 ","End":"02:36.430","Text":"We plot these as curves,"},{"Start":"02:36.430 ","End":"02:39.395","Text":"and then parameterized by t,"},{"Start":"02:39.395 ","End":"02:46.100","Text":"which typically goes all the way from minus infinity to infinity on the Cartesian plane."},{"Start":"02:46.100 ","End":"02:50.405","Text":"Solutions are also called trajectories or orbits."},{"Start":"02:50.405 ","End":"02:55.660","Text":"There\u0027s usually an infinite number of solutions for such a system of equations."},{"Start":"02:55.660 ","End":"02:58.460","Text":"Then I\u0027m going to give an example for"},{"Start":"02:58.460 ","End":"03:03.155","Text":"the particular system that we had above. Here\u0027s the sketch."},{"Start":"03:03.155 ","End":"03:06.210","Text":"One possible phase portrait is this."},{"Start":"03:06.210 ","End":"03:08.080","Text":"Is just a selection,"},{"Start":"03:08.080 ","End":"03:10.474","Text":"representative set of solutions."},{"Start":"03:10.474 ","End":"03:13.760","Text":"Obviously, there are solutions everywhere in the very dense,"},{"Start":"03:13.760 ","End":"03:20.209","Text":"but these are typical taken enough so to get the general idea."},{"Start":"03:20.209 ","End":"03:23.460","Text":"Now let\u0027s go into some more detail."}],"ID":10705},{"Watched":false,"Name":"Critical Point 1","Duration":"10m 28s","ChapterTopicVideoID":10356,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.945","Text":"Now, we come to a very central concept in such systems of differential equations,"},{"Start":"00:06.945 ","End":"00:09.180","Text":"that of a critical point."},{"Start":"00:09.180 ","End":"00:11.520","Text":"It has other names as we shall see,"},{"Start":"00:11.520 ","End":"00:17.415","Text":"it\u0027s also called a stationary point and it\u0027s also called an equilibrium solution."},{"Start":"00:17.415 ","End":"00:21.300","Text":"At the moment, we\u0027re going to apply the concept of"},{"Start":"00:21.300 ","End":"00:26.850","Text":"critical point to a system that we\u0027re studying x\u0027=Ax,"},{"Start":"00:26.850 ","End":"00:29.021","Text":"where A is the matrix,"},{"Start":"00:29.021 ","End":"00:32.090","Text":"but it can be generalized to non-linear systems."},{"Start":"00:32.090 ","End":"00:33.470","Text":"Never mind this notation,"},{"Start":"00:33.470 ","End":"00:37.285","Text":"this just means more general system."},{"Start":"00:37.285 ","End":"00:41.260","Text":"But let\u0027s see what it means in our case."},{"Start":"00:41.260 ","End":"00:46.580","Text":"We have a planar system 2D where x is x_1,"},{"Start":"00:46.580 ","End":"00:48.545","Text":"x_2, but in general,"},{"Start":"00:48.545 ","End":"00:52.910","Text":"a critical point is one where x\u0027 is 0,"},{"Start":"00:52.910 ","End":"00:58.310","Text":"the derivative of vector x with respect to t is 0."},{"Start":"00:58.310 ","End":"01:00.650","Text":"Component-wise, it means in our case,"},{"Start":"01:00.650 ","End":"01:03.905","Text":"that x\u0027_1 prime is 0 and x\u0027_2 is 0,"},{"Start":"01:03.905 ","End":"01:06.272","Text":"then it\u0027s going to be a critical point."},{"Start":"01:06.272 ","End":"01:14.780","Text":"Now, since x\u0027 is Ax and A is non-singular or determinant is not 0,"},{"Start":"01:14.780 ","End":"01:19.880","Text":"the only solution to such a system is the 0 vector,"},{"Start":"01:19.880 ","End":"01:21.710","Text":"or if you like the origin,"},{"Start":"01:21.710 ","End":"01:23.960","Text":"x is 0, 0."},{"Start":"01:23.960 ","End":"01:26.990","Text":"We only have one critical point and it\u0027s"},{"Start":"01:26.990 ","End":"01:31.550","Text":"always the origin for this particular kind of system."},{"Start":"01:31.550 ","End":"01:34.940","Text":"Now, we want to get to the sketching part."},{"Start":"01:34.940 ","End":"01:40.234","Text":"Well, the first part is sketching a direction field."},{"Start":"01:40.234 ","End":"01:43.370","Text":"You may or may not have studied direction fields,"},{"Start":"01:43.370 ","End":"01:46.130","Text":"I won\u0027t assume that you have."},{"Start":"01:46.130 ","End":"01:52.549","Text":"The sketching will most easily be done computer-aided."},{"Start":"01:52.549 ","End":"01:56.629","Text":"It\u0027s possible to do some sketching without computer,"},{"Start":"01:56.629 ","End":"01:59.210","Text":"but it\u0027s very labor-intensive,"},{"Start":"01:59.210 ","End":"02:00.605","Text":"meaning a lot of work,"},{"Start":"02:00.605 ","End":"02:02.120","Text":"a lot of computations,"},{"Start":"02:02.120 ","End":"02:05.990","Text":"so that\u0027s why we nowadays do things with the help of"},{"Start":"02:05.990 ","End":"02:11.495","Text":"a computer and tools like MATLAB and others."},{"Start":"02:11.495 ","End":"02:13.850","Text":"If we take, for example,"},{"Start":"02:13.850 ","End":"02:15.440","Text":"the system above,"},{"Start":"02:15.440 ","End":"02:16.685","Text":"it\u0027s already scrolled off,"},{"Start":"02:16.685 ","End":"02:20.930","Text":"I think A was 1, 2, 3, 0."},{"Start":"02:21.790 ","End":"02:26.090","Text":"Here it is, the direction field."},{"Start":"02:26.090 ","End":"02:34.190","Text":"What it means is that at each point we take x_1 and x_2."},{"Start":"02:34.190 ","End":"02:37.805","Text":"This axis would be the x_1 axis,"},{"Start":"02:37.805 ","End":"02:40.550","Text":"the vertical axis is the x_2 axis,"},{"Start":"02:40.550 ","End":"02:43.550","Text":"this is the origin."},{"Start":"02:43.550 ","End":"02:50.090","Text":"We can always compute A times x,"},{"Start":"02:50.090 ","End":"02:52.365","Text":"which is x_1, x_2,"},{"Start":"02:52.365 ","End":"02:55.310","Text":"and get a vector and then sketch it."},{"Start":"02:55.310 ","End":"02:58.250","Text":"You decide upon a scale for the arrows and so on,"},{"Start":"02:58.250 ","End":"03:01.024","Text":"and we do this for a bunch of points."},{"Start":"03:01.024 ","End":"03:05.590","Text":"We get a picture like this of the direction field."},{"Start":"03:05.590 ","End":"03:11.350","Text":"The next step is to draw some solutions,"},{"Start":"03:11.350 ","End":"03:14.240","Text":"are also called trajectories or orbits,"},{"Start":"03:14.240 ","End":"03:21.935","Text":"you just follow the arrows to the next arrow, something like this."},{"Start":"03:21.935 ","End":"03:25.970","Text":"It\u0027s fairly difficult to do,"},{"Start":"03:25.970 ","End":"03:29.120","Text":"which is why everything is then done with computers."},{"Start":"03:29.120 ","End":"03:32.210","Text":"I\u0027m just doing it freehand to give you an idea,"},{"Start":"03:32.210 ","End":"03:43.070","Text":"you just trace the arrows and you get something this,"},{"Start":"03:43.070 ","End":"03:46.590","Text":"this, this, this they\u0027re not supposed to intersect each other."},{"Start":"03:49.490 ","End":"03:52.127","Text":"Anyway, you get something like this."},{"Start":"03:52.127 ","End":"03:59.270","Text":"This is a very rough version of a phase portrait."},{"Start":"03:59.270 ","End":"04:03.350","Text":"We put arrows on each of these column trajectories,"},{"Start":"04:03.350 ","End":"04:04.910","Text":"like the arrows are going this way,"},{"Start":"04:04.910 ","End":"04:06.860","Text":"so this trajectory goes this way,"},{"Start":"04:06.860 ","End":"04:08.855","Text":"this trajectory goes this way,"},{"Start":"04:08.855 ","End":"04:10.387","Text":"this one\u0027s going this way,"},{"Start":"04:10.387 ","End":"04:12.485","Text":"you put an arrow on each one,"},{"Start":"04:12.485 ","End":"04:14.720","Text":"this one goes that way,"},{"Start":"04:14.720 ","End":"04:16.685","Text":"here we\u0027re going this way,"},{"Start":"04:16.685 ","End":"04:20.570","Text":"we\u0027re going this way, and so on."},{"Start":"04:20.570 ","End":"04:23.930","Text":"Here\u0027s a nicer picture that was,"},{"Start":"04:23.930 ","End":"04:26.390","Text":"again, the help of a computer,"},{"Start":"04:26.390 ","End":"04:32.180","Text":"but we still need to label each trajectory."},{"Start":"04:32.180 ","End":"04:35.265","Text":"Each one of these is a solution."},{"Start":"04:35.265 ","End":"04:41.585","Text":"As t varies, we go along the trajectory from minus infinity to infinity."},{"Start":"04:41.585 ","End":"04:44.060","Text":"We go along a trajectory."},{"Start":"04:44.060 ","End":"04:46.820","Text":"I\u0027ll put a few more here, here, here,"},{"Start":"04:46.820 ","End":"04:48.470","Text":"let\u0027s say here we are going this way,"},{"Start":"04:48.470 ","End":"04:50.565","Text":"we\u0027re going this way."},{"Start":"04:50.565 ","End":"04:54.010","Text":"There are actually, let\u0027s see,"},{"Start":"04:54.010 ","End":"04:56.930","Text":"even 5 more solutions,"},{"Start":"04:56.930 ","End":"05:01.325","Text":"particular ones, the origin itself is a solution,"},{"Start":"05:01.325 ","End":"05:04.895","Text":"but it stays at the same place, a stationary point."},{"Start":"05:04.895 ","End":"05:06.387","Text":"In other words,"},{"Start":"05:06.387 ","End":"05:09.425","Text":"x(t) is the origin for every t,"},{"Start":"05:09.425 ","End":"05:11.420","Text":"there are also 4 more,"},{"Start":"05:11.420 ","End":"05:14.790","Text":"which are these straight lines, in this case."},{"Start":"05:15.040 ","End":"05:21.020","Text":"The arrows I see here go this way, this here,"},{"Start":"05:21.020 ","End":"05:23.615","Text":"it goes this way, here,"},{"Start":"05:23.615 ","End":"05:26.960","Text":"this way, and here, this way."},{"Start":"05:26.960 ","End":"05:34.355","Text":"Note that all the orbits go away from the origin,"},{"Start":"05:34.355 ","End":"05:36.845","Text":"with the exception of 2 orbits,"},{"Start":"05:36.845 ","End":"05:38.629","Text":"which is this orbit,"},{"Start":"05:38.629 ","End":"05:40.140","Text":"which is a half-line,"},{"Start":"05:40.140 ","End":"05:41.595","Text":"not including the origin,"},{"Start":"05:41.595 ","End":"05:44.870","Text":"and this one here goes towards the origin, these two go away."},{"Start":"05:44.870 ","End":"05:49.460","Text":"Well, pretty much they all go away and we\u0027ll talk about that in a moment,"},{"Start":"05:49.460 ","End":"05:54.420","Text":"that will mean that this critical point is unstable."},{"Start":"05:54.890 ","End":"05:56.930","Text":"The shape of this,"},{"Start":"05:56.930 ","End":"05:59.345","Text":"it\u0027s actually called a saddle point,"},{"Start":"05:59.345 ","End":"06:01.490","Text":"maybe looked like a sagittal."},{"Start":"06:01.490 ","End":"06:04.470","Text":"Anyway, let\u0027s continue."},{"Start":"06:04.690 ","End":"06:13.740","Text":"Our main goal here is to classify the critical points and to sketch."},{"Start":"06:14.210 ","End":"06:16.590","Text":"There are 2 typical tasks,"},{"Start":"06:16.590 ","End":"06:19.560","Text":"there is actually 3 tasks."},{"Start":"06:19.560 ","End":"06:23.840","Text":"I call it task 0 is to find the general solution for the system,"},{"Start":"06:23.840 ","End":"06:25.520","Text":"but that belongs in another chapter,"},{"Start":"06:25.520 ","End":"06:28.460","Text":"so I\u0027m just assuming that we can do that."},{"Start":"06:28.460 ","End":"06:31.207","Text":"There are 2 other tasks,"},{"Start":"06:31.207 ","End":"06:34.855","Text":"one of them is to classify the critical points."},{"Start":"06:34.855 ","End":"06:38.260","Text":"It turns out that there are 2 things you can say about a critical point."},{"Start":"06:38.260 ","End":"06:39.864","Text":"What is its type?"},{"Start":"06:39.864 ","End":"06:41.500","Text":"The type, in this case,"},{"Start":"06:41.500 ","End":"06:43.555","Text":"will be saddle point,"},{"Start":"06:43.555 ","End":"06:46.690","Text":"we\u0027ll see, and also the stability."},{"Start":"06:46.690 ","End":"06:48.460","Text":"I gave you a hint at what that means,"},{"Start":"06:48.460 ","End":"06:56.965","Text":"it means whether the solutions are leading away from the origin or,"},{"Start":"06:56.965 ","End":"07:00.100","Text":"we\u0027ll see what other possibilities there are."},{"Start":"07:00.570 ","End":"07:03.400","Text":"Once we\u0027ve classified the critical point,"},{"Start":"07:03.400 ","End":"07:06.865","Text":"we can draw a very rough sketch,"},{"Start":"07:06.865 ","End":"07:10.780","Text":"a phase plane portrait just as a general idea,"},{"Start":"07:10.780 ","End":"07:13.237","Text":"but it won\u0027t be precise."},{"Start":"07:13.237 ","End":"07:17.019","Text":"Like in our example where the matrix was this,"},{"Start":"07:17.019 ","End":"07:22.615","Text":"you can compute the eigenvalues and see that they are minus 2 and 3,"},{"Start":"07:22.615 ","End":"07:29.150","Text":"and as we shall show when you have 2 real eigenvalues of opposite signs,"},{"Start":"07:29.150 ","End":"07:32.360","Text":"it means that we have a saddle point which is always"},{"Start":"07:32.360 ","End":"07:36.410","Text":"unstable and you get the general shape."},{"Start":"07:36.410 ","End":"07:39.065","Text":"But there\u0027s no precision there."},{"Start":"07:39.065 ","End":"07:41.630","Text":"If you want any kind of precision,"},{"Start":"07:41.630 ","End":"07:43.490","Text":"we have to do some computations."},{"Start":"07:43.490 ","End":"07:49.625","Text":"The first step is to compute the eigenvalues of the matrix."},{"Start":"07:49.625 ","End":"07:52.910","Text":"Usually, the eigenvalues will tell us what we need to know,"},{"Start":"07:52.910 ","End":"07:57.155","Text":"sometimes we also need the eigenvectors,"},{"Start":"07:57.155 ","End":"07:59.045","Text":"and when we have those,"},{"Start":"07:59.045 ","End":"08:05.520","Text":"then we can consult the lookup table or map that I\u0027m about to show."},{"Start":"08:05.650 ","End":"08:09.620","Text":"Mostly, it\u0027s determined according to eigenvalues,"},{"Start":"08:09.620 ","End":"08:13.640","Text":"but sometimes we need the eigenvectors also."},{"Start":"08:13.640 ","End":"08:16.475","Text":"Here\u0027s the classification."},{"Start":"08:16.475 ","End":"08:20.555","Text":"The 2 eigenvalues might both be real,"},{"Start":"08:20.555 ","End":"08:24.065","Text":"and that\u0027s part a, or they might be complex."},{"Start":"08:24.065 ","End":"08:25.708","Text":"If they\u0027re real,"},{"Start":"08:25.708 ","End":"08:28.460","Text":"they could be distinct,"},{"Start":"08:28.460 ","End":"08:31.460","Text":"2 different real eigenvalues,"},{"Start":"08:31.460 ","End":"08:32.780","Text":"or they could be repeated,"},{"Start":"08:32.780 ","End":"08:35.591","Text":"the 2 eigenvalues are the same."},{"Start":"08:35.591 ","End":"08:39.170","Text":"If they\u0027re distinct, they could have opposite signs,"},{"Start":"08:39.170 ","End":"08:40.700","Text":"meaning one positive, one negative,"},{"Start":"08:40.700 ","End":"08:42.815","Text":"or both positive or both negative."},{"Start":"08:42.815 ","End":"08:46.040","Text":"Then in the table, in this color,"},{"Start":"08:46.040 ","End":"08:50.495","Text":"I wrote the type of node and here I wrote the stability."},{"Start":"08:50.495 ","End":"08:55.610","Text":"At the moment we just take these as labels and not worry too much what they mean."},{"Start":"08:55.610 ","End":"08:59.630","Text":"The types are just reminiscent from the pictures,"},{"Start":"08:59.630 ","End":"09:01.445","Text":"that\u0027s what they\u0027re called."},{"Start":"09:01.445 ","End":"09:04.850","Text":"We had, in our case,"},{"Start":"09:04.850 ","End":"09:07.490","Text":"if you check the eigenvalues for our matrix,"},{"Start":"09:07.490 ","End":"09:10.640","Text":"you\u0027ll find that one is positive and one is negative,"},{"Start":"09:10.640 ","End":"09:15.530","Text":"and that would tell us that it is indeed a saddle point and it\u0027s unstable."},{"Start":"09:15.530 ","End":"09:19.730","Text":"I\u0027m not going to go into every case here,"},{"Start":"09:19.730 ","End":"09:21.605","Text":"but you should have this table."},{"Start":"09:21.605 ","End":"09:27.155","Text":"Just notice that if we have a repeated eigenvalue,"},{"Start":"09:27.155 ","End":"09:30.875","Text":"then we also have to look at the eigenvectors,"},{"Start":"09:30.875 ","End":"09:35.120","Text":"or at least just to know if there\u0027s 1 eigenvector or 2 eigenvectors,"},{"Start":"09:35.120 ","End":"09:40.265","Text":"that could be the case when we have a repeated eigenvalue,"},{"Start":"09:40.265 ","End":"09:43.150","Text":"there could be 1 or 2."},{"Start":"09:43.660 ","End":"09:49.850","Text":"Then the complex cases,"},{"Start":"09:49.850 ","End":"09:51.800","Text":"it could be pure imaginary,"},{"Start":"09:51.800 ","End":"09:53.690","Text":"meaning the real part is 0,"},{"Start":"09:53.690 ","End":"09:57.953","Text":"it\u0027s just got the i part."},{"Start":"09:57.953 ","End":"10:03.025","Text":"It\u0027s going to be a + or - bi."},{"Start":"10:03.025 ","End":"10:06.095","Text":"If a is 0, it\u0027s just imaginary."},{"Start":"10:06.095 ","End":"10:11.155","Text":"Then the type is called a center and it\u0027s stable."},{"Start":"10:11.155 ","End":"10:14.210","Text":"Then if a is positive,"},{"Start":"10:14.210 ","End":"10:17.900","Text":"we have a spiral point which is unstable,"},{"Start":"10:17.900 ","End":"10:20.195","Text":"and if a is negative still a spiral point,"},{"Start":"10:20.195 ","End":"10:22.865","Text":"but something called asymptotically stable."},{"Start":"10:22.865 ","End":"10:25.565","Text":"The asymp means asymptotically."},{"Start":"10:25.565 ","End":"10:29.490","Text":"Let\u0027s talk about this concept of stability."}],"ID":10706},{"Watched":false,"Name":"Critical Point 2","Duration":"10m 25s","ChapterTopicVideoID":10357,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.140 ","End":"00:04.975","Text":"Now about stability, what can we say?"},{"Start":"00:04.975 ","End":"00:08.320","Text":"I want to remind you that the critical point, in our case,"},{"Start":"00:08.320 ","End":"00:11.440","Text":"it\u0027s just x=0,"},{"Start":"00:11.440 ","End":"00:14.750","Text":"is also an equilibrium solution."},{"Start":"00:14.790 ","End":"00:19.000","Text":"If we take x(t)=0 for all t,"},{"Start":"00:19.000 ","End":"00:25.285","Text":"then it\u0027s a constant solution or equilibrium and it\u0027s the origin in our case."},{"Start":"00:25.285 ","End":"00:28.030","Text":"What we\u0027re concerned with, in contrast,"},{"Start":"00:28.030 ","End":"00:30.130","Text":"is the other solutions besides"},{"Start":"00:30.130 ","End":"00:36.565","Text":"the equilibrium solution and how they behave as t goes to infinity."},{"Start":"00:36.565 ","End":"00:39.670","Text":"I\u0027ll relate to the word most in a moment."},{"Start":"00:39.670 ","End":"00:42.985","Text":"There\u0027s generally 3 cases."},{"Start":"00:42.985 ","End":"00:48.935","Text":"If the solution is converged to the critical point as t goes to infinity,"},{"Start":"00:48.935 ","End":"00:53.404","Text":"of course, then we call that asymptotically stable."},{"Start":"00:53.404 ","End":"00:57.665","Text":"If they move away from the critical point to infinity,"},{"Start":"00:57.665 ","End":"01:01.625","Text":"then we call that unstable."},{"Start":"01:01.625 ","End":"01:08.360","Text":"If they stay in a fixed orbit in some bounded range around the critical point,"},{"Start":"01:08.360 ","End":"01:11.540","Text":"and they don\u0027t go to infinity and they don\u0027t go towards the critical point,"},{"Start":"01:11.540 ","End":"01:16.910","Text":"then that\u0027s called stable or sometimes neutrally stable."},{"Start":"01:16.910 ","End":"01:19.700","Text":"Now about the word most,"},{"Start":"01:19.700 ","End":"01:22.670","Text":"that\u0027s because in our example,"},{"Start":"01:22.670 ","End":"01:24.830","Text":"like in the saddle point,"},{"Start":"01:24.830 ","End":"01:28.985","Text":"all but 2 of the solutions moved away from the origin."},{"Start":"01:28.985 ","End":"01:36.170","Text":"I mean infinity of them except for 2, we can say that\u0027s most."},{"Start":"01:36.170 ","End":"01:39.695","Text":"A little bit of finite number and then it\u0027s unstable."},{"Start":"01:39.695 ","End":"01:43.620","Text":"Saddle point is always unstable."},{"Start":"01:43.660 ","End":"01:47.010","Text":"I just wanted to give you an introduction to these concepts."},{"Start":"01:47.010 ","End":"01:51.520","Text":"We\u0027re not going to go in very deeply into these."},{"Start":"01:51.520 ","End":"01:54.420","Text":"Now the fun part."},{"Start":"01:54.420 ","End":"02:00.485","Text":"I\u0027m just going to familiarize you with some of the types of critical points that we have."},{"Start":"02:00.485 ","End":"02:06.035","Text":"We already are familiar with the saddle point which looked something like this,"},{"Start":"02:06.035 ","End":"02:13.970","Text":"where pretty much everything moved away from the origin, so it\u0027s unstable."},{"Start":"02:13.970 ","End":"02:15.650","Text":"There were just 2 particular ones,"},{"Start":"02:15.650 ","End":"02:21.260","Text":"this one and this one where the solutions went towards the origin."},{"Start":"02:21.260 ","End":"02:23.945","Text":"Then there\u0027s something called a node."},{"Start":"02:23.945 ","End":"02:28.370","Text":"Just get the general impression of the picture."},{"Start":"02:28.370 ","End":"02:35.794","Text":"The orbits, they all go towards the critical point."},{"Start":"02:35.794 ","End":"02:37.910","Text":"This one would be asymptotically"},{"Start":"02:37.910 ","End":"02:41.105","Text":"stable because at infinity it all leads to the critical point,"},{"Start":"02:41.105 ","End":"02:46.850","Text":"but the arrows could be reversed and then it would be an unstable node."},{"Start":"02:46.850 ","End":"02:50.660","Text":"There\u0027s something called an improper node,"},{"Start":"02:50.660 ","End":"02:54.780","Text":"which looks somewhat like this,"},{"Start":"02:55.160 ","End":"02:59.325","Text":"or it could be the other way."},{"Start":"02:59.325 ","End":"03:03.260","Text":"One is like an S and one\u0027s like a Z, if you like."},{"Start":"03:03.260 ","End":"03:06.830","Text":"Both of these are asymptotically stable."},{"Start":"03:06.830 ","End":"03:10.760","Text":"The solutions go towards the critical point which is the origin."},{"Start":"03:10.760 ","End":"03:13.115","Text":"If I reverse the arrows on these,"},{"Start":"03:13.115 ","End":"03:19.180","Text":"then they are unstable improper nodes."},{"Start":"03:19.180 ","End":"03:21.335","Text":"There\u0027s more coming up."},{"Start":"03:21.335 ","End":"03:24.230","Text":"Let\u0027s see what else we have."},{"Start":"03:24.230 ","End":"03:26.929","Text":"We have a proper node,"},{"Start":"03:26.929 ","End":"03:29.735","Text":"also known as a star point."},{"Start":"03:29.735 ","End":"03:37.014","Text":"Just what it looks like. Everything\u0027s straight lines leading towards the origin,"},{"Start":"03:37.014 ","End":"03:38.870","Text":"but it could be the other way around."},{"Start":"03:38.870 ","End":"03:41.030","Text":"If we reverse all the arrows,"},{"Start":"03:41.030 ","End":"03:42.710","Text":"that\u0027s also a proper node."},{"Start":"03:42.710 ","End":"03:44.570","Text":"In one case it\u0027s asymptotically"},{"Start":"03:44.570 ","End":"03:50.350","Text":"stable and in the other case when the arrows are going outwards, it\u0027s unstable."},{"Start":"03:50.350 ","End":"03:52.720","Text":"There\u0027s something called a center point."},{"Start":"03:52.720 ","End":"03:56.365","Text":"They don\u0027t go to infinity and they don\u0027t go towards the critical point."},{"Start":"03:56.365 ","End":"03:59.600","Text":"Just going round and round."},{"Start":"03:59.610 ","End":"04:03.985","Text":"Sometimes we want to know if it\u0027s counterclockwise or clockwise."},{"Start":"04:03.985 ","End":"04:07.315","Text":"There are simple ways of testing that."},{"Start":"04:07.315 ","End":"04:09.984","Text":"There\u0027s something called the spiral point."},{"Start":"04:09.984 ","End":"04:11.695","Text":"Looks like this."},{"Start":"04:11.695 ","End":"04:13.460","Text":"In this one,"},{"Start":"04:13.460 ","End":"04:18.970","Text":"this is going to be unstable because they\u0027re going further away to infinity."},{"Start":"04:18.970 ","End":"04:22.225","Text":"But if the arrows are reversed, it\u0027s asymptotically stable."},{"Start":"04:22.225 ","End":"04:28.960","Text":"I guess this one would be an asymptotically stable spiral point."},{"Start":"04:30.230 ","End":"04:36.820","Text":"That\u0027s the familiarization generally with what types of critical points we have."},{"Start":"04:36.820 ","End":"04:39.760","Text":"Now let\u0027s do a couple of solved examples."},{"Start":"04:39.760 ","End":"04:44.140","Text":"In this example, we are given a system in"},{"Start":"04:44.140 ","End":"04:49.975","Text":"this form and we want to classify the critical point and sketch a phase portrait."},{"Start":"04:49.975 ","End":"04:55.090","Text":"We might even do a bit more and actually give a general solution for the system."},{"Start":"04:55.090 ","End":"05:01.255","Text":"We can rewrite the system in matrix form,"},{"Start":"05:01.255 ","End":"05:07.380","Text":"where this matrix A would just be diagonal matrix 2, 0, 0, 2."},{"Start":"05:07.380 ","End":"05:15.115","Text":"It\u0027s diagonal because here it\u0027s 2X_1 plus 0X_2 and here it\u0027s 0X_1 plus 2X_2."},{"Start":"05:15.115 ","End":"05:18.718","Text":"So this is what we get."},{"Start":"05:18.718 ","End":"05:20.180","Text":"For this matrix, well,"},{"Start":"05:20.180 ","End":"05:23.860","Text":"you can see because it\u0027s a diagonal matrix these two are the eigenvalues,"},{"Start":"05:23.860 ","End":"05:26.620","Text":"so it\u0027s a repeated eigenvalue,"},{"Start":"05:26.620 ","End":"05:28.848","Text":"2 is repeated,"},{"Start":"05:28.848 ","End":"05:32.540","Text":"and it happens to have 2 eigenvectors."},{"Start":"05:32.540 ","End":"05:35.620","Text":"You might get a different answer if you do it."},{"Start":"05:35.620 ","End":"05:37.835","Text":"There\u0027s more than one way of doing it."},{"Start":"05:37.835 ","End":"05:40.070","Text":"So that\u0027s the point."},{"Start":"05:40.070 ","End":"05:43.925","Text":"We have a repeated eigenvalue and 2 eigenvectors."},{"Start":"05:43.925 ","End":"05:46.580","Text":"We go and look in the table."},{"Start":"05:46.580 ","End":"05:48.710","Text":"I\u0027m not going to scroll back,"},{"Start":"05:48.710 ","End":"05:51.700","Text":"but if you look in the table when we have this situation,"},{"Start":"05:51.700 ","End":"05:55.570","Text":"then we have an unstable proper node."},{"Start":"05:55.570 ","End":"05:58.670","Text":"The proper node is also called a star point."},{"Start":"05:58.670 ","End":"06:02.390","Text":"It\u0027s always at the origin for this kind of system."},{"Start":"06:02.390 ","End":"06:06.545","Text":"Just want to dig a bit deeper to explain the concepts."},{"Start":"06:06.545 ","End":"06:09.365","Text":"From here, we can get the general solution."},{"Start":"06:09.365 ","End":"06:18.350","Text":"Because the general solution is e^ Lambda t for the first eigenvalue,"},{"Start":"06:18.350 ","End":"06:20.075","Text":"which is e^2t,"},{"Start":"06:20.075 ","End":"06:23.490","Text":"times the first eigenvector."},{"Start":"06:23.490 ","End":"06:26.430","Text":"Similarly for the other one on the other eigenvector."},{"Start":"06:26.430 ","End":"06:30.920","Text":"We put 2 arbitrary constants, one for each."},{"Start":"06:30.920 ","End":"06:35.390","Text":"Now because it\u0027s the same eigenvalue,"},{"Start":"06:35.390 ","End":"06:41.115","Text":"we can simplify a bit and write it as e^2t."},{"Start":"06:41.115 ","End":"06:42.620","Text":"Then if I multiply this out,"},{"Start":"06:42.620 ","End":"06:44.570","Text":"I\u0027ve got a vector C_1, C_2."},{"Start":"06:44.570 ","End":"06:47.780","Text":"But since C_1, C_2 are arbitrary,"},{"Start":"06:47.780 ","End":"06:58.830","Text":"it\u0027s just e^2t times any vector C. That gives us the star shape,"},{"Start":"06:58.830 ","End":"07:01.520","Text":"but not the star shape from before."},{"Start":"07:01.520 ","End":"07:02.840","Text":"I just copied the picture,"},{"Start":"07:02.840 ","End":"07:07.475","Text":"but we need to reverse the direction of the arrows."},{"Start":"07:07.475 ","End":"07:09.230","Text":"I\u0027ll explain a little bit more in a moment."},{"Start":"07:09.230 ","End":"07:11.525","Text":"Let me just highlight those."},{"Start":"07:11.525 ","End":"07:15.370","Text":"As we said, an unstable proper node,"},{"Start":"07:15.370 ","End":"07:17.970","Text":"because C could be any vector."},{"Start":"07:17.970 ","End":"07:21.530","Text":"If we take any vector and multiply it by e^2t,"},{"Start":"07:21.530 ","End":"07:24.950","Text":"when t goes from minus infinity to infinity,"},{"Start":"07:24.950 ","End":"07:27.465","Text":"e^2t goes from 0 to infinity."},{"Start":"07:27.465 ","End":"07:28.745","Text":"Any vector you take,"},{"Start":"07:28.745 ","End":"07:31.910","Text":"we can shorten it to 0 or lengthen it to infinity."},{"Start":"07:31.910 ","End":"07:36.580","Text":"That gives the spokes of this star, if you like."},{"Start":"07:36.580 ","End":"07:39.835","Text":"That\u0027s the explanation of why this is a star."},{"Start":"07:39.835 ","End":"07:41.580","Text":"Got carried away with the talking."},{"Start":"07:41.580 ","End":"07:46.485","Text":"Here\u0027s the writing. Here I\u0027ve just written what I already said."},{"Start":"07:46.485 ","End":"07:51.770","Text":"Now for the computer drawn version of this,"},{"Start":"07:51.770 ","End":"07:58.130","Text":"here it is, but I forgot to put the arrows."},{"Start":"07:58.130 ","End":"08:00.230","Text":"Notice that all the arrows are going outwards,"},{"Start":"08:00.230 ","End":"08:02.789","Text":"so we would put an arrow."},{"Start":"08:03.580 ","End":"08:08.420","Text":"It\u0027s up to you how many solutions you draw."},{"Start":"08:08.420 ","End":"08:10.760","Text":"I think this is just about a representative set of"},{"Start":"08:10.760 ","End":"08:14.300","Text":"solutions and gives us the idea what it looks like."},{"Start":"08:14.300 ","End":"08:16.430","Text":"This would be a phase portrait."},{"Start":"08:16.430 ","End":"08:19.384","Text":"Let\u0027s move on to the next example."},{"Start":"08:19.384 ","End":"08:23.060","Text":"Here we are with the second example."},{"Start":"08:23.060 ","End":"08:30.145","Text":"We also have to classify the critical point at the origin and sketch a phase portrait."},{"Start":"08:30.145 ","End":"08:32.210","Text":"As in the first example,"},{"Start":"08:32.210 ","End":"08:34.655","Text":"we\u0027ll write it in matrix form."},{"Start":"08:34.655 ","End":"08:37.010","Text":"This is what we would get."},{"Start":"08:37.010 ","End":"08:43.805","Text":"0X_1 plus 3X_2 and minus X_1 plus 0X_2 from here."},{"Start":"08:43.805 ","End":"08:53.655","Text":"The eigenvalues come out to be plus or minus root 3 times i, both imaginary."},{"Start":"08:53.655 ","End":"08:55.400","Text":"If we look at the table,"},{"Start":"08:55.400 ","End":"08:59.515","Text":"then we know that we have a stable center point."},{"Start":"08:59.515 ","End":"09:03.317","Text":"Usually we want to find out,"},{"Start":"09:03.317 ","End":"09:07.530","Text":"in this case, whether it\u0027s clockwise or counterclockwise."},{"Start":"09:07.600 ","End":"09:10.700","Text":"Here\u0027s a picture of a center point,"},{"Start":"09:10.700 ","End":"09:13.700","Text":"but we don\u0027t know if the arrows are correct or not."},{"Start":"09:13.700 ","End":"09:17.090","Text":"What we often do is take a point like 1,"},{"Start":"09:17.090 ","End":"09:20.460","Text":"0 and we can compute X\u0027 from 1,"},{"Start":"09:20.460 ","End":"09:23.780","Text":"0 by just multiplying it by matrix A."},{"Start":"09:23.780 ","End":"09:26.255","Text":"Here\u0027s matrix A, here is 1, 0."},{"Start":"09:26.255 ","End":"09:27.740","Text":"We multiply it out,"},{"Start":"09:27.740 ","End":"09:29.855","Text":"we get 0 minus 1."},{"Start":"09:29.855 ","End":"09:32.975","Text":"The second component is minus 1,"},{"Start":"09:32.975 ","End":"09:35.405","Text":"which means it\u0027s downwards."},{"Start":"09:35.405 ","End":"09:38.300","Text":"If I take the point 1, 0 and going downwards."},{"Start":"09:38.300 ","End":"09:41.555","Text":"Actually this one is wrong, gets clockwise."},{"Start":"09:41.555 ","End":"09:46.010","Text":"We need to put the arrows this way and let\u0027s just ignore these arrows."},{"Start":"09:46.010 ","End":"09:47.930","Text":"It\u0027s this way."},{"Start":"09:47.930 ","End":"09:55.310","Text":"Finally, I\u0027ll show you what a computer drawn sketch might look like. Here it is."},{"Start":"09:55.310 ","End":"10:00.000","Text":"Here\u0027s one I made on one of these websites with calculators."},{"Start":"10:00.040 ","End":"10:06.050","Text":"We just have to make sure that we also add the arrows, which is customary."},{"Start":"10:06.050 ","End":"10:09.275","Text":"Now we said clockwise, so here,"},{"Start":"10:09.275 ","End":"10:13.160","Text":"here, here, here, and here."},{"Start":"10:13.160 ","End":"10:17.900","Text":"Or we could also put them here and so on. That\u0027s about it."},{"Start":"10:17.900 ","End":"10:21.665","Text":"There are solved examples following the tutorial,"},{"Start":"10:21.665 ","End":"10:25.140","Text":"so I\u0027ll leave the tutorial at that."}],"ID":10707},{"Watched":false,"Name":"Phase portrait of Non Linear Systems","Duration":"13m 21s","ChapterTopicVideoID":10358,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.310","Text":"Starting a new topic,"},{"Start":"00:02.310 ","End":"00:08.310","Text":"we studied phase plane and phase portraits of linear systems of a certain kind,"},{"Start":"00:08.310 ","End":"00:13.455","Text":"homogeneous constant coefficients, 2 dimensional."},{"Start":"00:13.455 ","End":"00:16.560","Text":"Now, we\u0027re going to generalize a bit."},{"Start":"00:16.560 ","End":"00:19.722","Text":"We\u0027re still going to stay in 2 dimensions."},{"Start":"00:19.722 ","End":"00:21.945","Text":"That\u0027s the planar."},{"Start":"00:21.945 ","End":"00:27.345","Text":"We\u0027re going to study a certain kind of non-linear system called autonomous."},{"Start":"00:27.345 ","End":"00:29.625","Text":"A point on notation,"},{"Start":"00:29.625 ","End":"00:33.540","Text":"here in this clip we\u0027ll prefer to use x and"},{"Start":"00:33.540 ","End":"00:39.165","Text":"y as components of a vector in the plane rather than x_1, x_2."},{"Start":"00:39.165 ","End":"00:42.910","Text":"I don\u0027t know why. It\u0027s just more commonly seen this way."},{"Start":"00:42.980 ","End":"00:47.110","Text":"The autonomous system looks like this."},{"Start":"00:47.110 ","End":"00:51.100","Text":"We have x\u0027 is some function of x and y,"},{"Start":"00:51.100 ","End":"00:55.225","Text":"and y\u0027 is some function of x and y."},{"Start":"00:55.225 ","End":"00:57.010","Text":"The thing to notice,"},{"Start":"00:57.010 ","End":"00:58.540","Text":"and that\u0027s what makes it autonomous,"},{"Start":"00:58.540 ","End":"01:02.620","Text":"is that these functions are not dependent on t. T"},{"Start":"01:02.620 ","End":"01:07.845","Text":"is missing in the equations or on the right-hand side,"},{"Start":"01:07.845 ","End":"01:09.955","Text":"so it\u0027s not time-dependent."},{"Start":"01:09.955 ","End":"01:15.595","Text":"For one thing, it means that at any given point in the plane, and if any x, y,"},{"Start":"01:15.595 ","End":"01:19.345","Text":"we can compute the direction vector,"},{"Start":"01:19.345 ","End":"01:25.700","Text":"and we could draw the direction field because it doesn\u0027t depend on time."},{"Start":"01:25.730 ","End":"01:29.970","Text":"As an example, the previous system we studied,"},{"Start":"01:29.970 ","End":"01:32.415","Text":"which was this,"},{"Start":"01:32.415 ","End":"01:33.960","Text":"there we had x_1, x_2,"},{"Start":"01:33.960 ","End":"01:35.473","Text":"here we have x, y."},{"Start":"01:35.473 ","End":"01:40.460","Text":"The derivative, is some matrix times the vector itself,"},{"Start":"01:40.460 ","End":"01:41.960","Text":"and this constant coefficients."},{"Start":"01:41.960 ","End":"01:45.949","Text":"We can actually write this in this form by expanding,"},{"Start":"01:45.949 ","End":"01:49.085","Text":"and we get x\u0027 is ax plus by,"},{"Start":"01:49.085 ","End":"01:51.215","Text":"and y\u0027 is cx plus dy,"},{"Start":"01:51.215 ","End":"01:53.420","Text":"which certainly is of the form here."},{"Start":"01:53.420 ","End":"01:55.370","Text":"This is a function of x and y,"},{"Start":"01:55.370 ","End":"01:57.260","Text":"and this is another function of x and y,"},{"Start":"01:57.260 ","End":"01:58.955","Text":"and t doesn\u0027t appear."},{"Start":"01:58.955 ","End":"02:02.660","Text":"Just to give it as an example and to show that the previous kind"},{"Start":"02:02.660 ","End":"02:06.305","Text":"we\u0027ve studied is of this type, so we\u0027re generalizing."},{"Start":"02:06.305 ","End":"02:10.580","Text":"Just as before, we\u0027re going to be concerning ourselves with critical points,"},{"Start":"02:10.580 ","End":"02:12.320","Text":"but there will be differences."},{"Start":"02:12.320 ","End":"02:18.123","Text":"For example, previously a critical point was always at the origin,"},{"Start":"02:18.123 ","End":"02:22.460","Text":"but in the non-linear system that we\u0027re studying,"},{"Start":"02:22.460 ","End":"02:25.760","Text":"we could have any number of critical points."},{"Start":"02:25.760 ","End":"02:28.430","Text":"It could be 0, could be more than 1,"},{"Start":"02:28.430 ","End":"02:32.255","Text":"could be 1, depends on the system."},{"Start":"02:32.255 ","End":"02:35.090","Text":"The way we find them is the same as before."},{"Start":"02:35.090 ","End":"02:38.855","Text":"We just set each component of the derivative to be 0."},{"Start":"02:38.855 ","End":"02:41.930","Text":"In this case, x\u0027 and y\u0027 both 0."},{"Start":"02:41.930 ","End":"02:44.270","Text":"What it comes down to, if you look here,"},{"Start":"02:44.270 ","End":"02:48.375","Text":"is we want both F and G to be 0."},{"Start":"02:48.375 ","End":"02:52.990","Text":"We get 2 equations and 2 unknowns,"},{"Start":"02:52.990 ","End":"02:55.280","Text":"which may or may not be easy to solve."},{"Start":"02:55.280 ","End":"03:00.500","Text":"In fact, it might be impossible to solve analytically, maybe only numerically."},{"Start":"03:00.500 ","End":"03:02.060","Text":"But in our cases,"},{"Start":"03:02.060 ","End":"03:05.150","Text":"we\u0027re going to have cases which we can solve algebraically"},{"Start":"03:05.150 ","End":"03:08.355","Text":"fairly simply. I just wrote that here."},{"Start":"03:08.355 ","End":"03:10.835","Text":"In general, you may not be able to solve it,"},{"Start":"03:10.835 ","End":"03:12.440","Text":"but it\u0027s going to be, I\u0027m not sure,"},{"Start":"03:12.440 ","End":"03:14.844","Text":"if it\u0027s solvable or soluble,"},{"Start":"03:14.844 ","End":"03:17.255","Text":"whatever the right word is."},{"Start":"03:17.255 ","End":"03:20.765","Text":"Anyway, we\u0027re only going to consider cases that we can solve."},{"Start":"03:20.765 ","End":"03:24.290","Text":"The problems are more involved and the solutions are going to be"},{"Start":"03:24.290 ","End":"03:27.410","Text":"more complicated solutions, trajectories, orbits."},{"Start":"03:27.410 ","End":"03:32.045","Text":"This time they could be influenced by more than 1 critical point."},{"Start":"03:32.045 ","End":"03:35.000","Text":"I\u0027m just talking generally,"},{"Start":"03:35.000 ","End":"03:38.300","Text":"the phase portrait is usually more involved,"},{"Start":"03:38.300 ","End":"03:39.950","Text":"and messy, chaotic,"},{"Start":"03:39.950 ","End":"03:42.410","Text":"whatever you want to call it."},{"Start":"03:42.410 ","End":"03:45.575","Text":"The main technique we use,"},{"Start":"03:45.575 ","End":"03:48.290","Text":"it\u0027s going to be called linearization,"},{"Start":"03:48.290 ","End":"03:56.750","Text":"is to take some approximation of"},{"Start":"03:56.750 ","End":"04:05.540","Text":"the behavior of the system near critical points that will make it like a linear system,"},{"Start":"04:05.540 ","End":"04:09.335","Text":"linear planar constant coefficients homogeneous,"},{"Start":"04:09.335 ","End":"04:13.235","Text":"and that process is called linearization."},{"Start":"04:13.235 ","End":"04:21.515","Text":"Linearization is replacing the function by a linear approximation,"},{"Start":"04:21.515 ","End":"04:25.069","Text":"and we do that near each critical point."},{"Start":"04:25.069 ","End":"04:29.695","Text":"That linearization is sometimes called tangent plane approximation,"},{"Start":"04:29.695 ","End":"04:36.733","Text":"and we do this separately for F and for G at each critical point."},{"Start":"04:36.733 ","End":"04:39.300","Text":"Let\u0027s say, the critical point is a,"},{"Start":"04:39.300 ","End":"04:41.230","Text":"b, one of them."},{"Start":"04:41.230 ","End":"04:43.820","Text":"I\u0027m going to be talking a bit abstractly,"},{"Start":"04:43.820 ","End":"04:46.580","Text":"but once you see an example it\u0027ll be clearer,"},{"Start":"04:46.580 ","End":"04:49.070","Text":"then you can go back and re-read,"},{"Start":"04:49.070 ","End":"04:51.440","Text":"and you\u0027ll see it makes more sense."},{"Start":"04:51.440 ","End":"04:58.445","Text":"Anyway, after we get this linear tangent plane approximation,"},{"Start":"04:58.445 ","End":"05:06.080","Text":"we shift so that the critical point becomes the origin."},{"Start":"05:06.080 ","End":"05:08.150","Text":"After we do that,"},{"Start":"05:08.150 ","End":"05:11.300","Text":"we then get a linear system just like this,"},{"Start":"05:11.300 ","End":"05:12.860","Text":"which we used to have,"},{"Start":"05:12.860 ","End":"05:19.840","Text":"where it turns out that the matrix A at the point a,"},{"Start":"05:19.840 ","End":"05:28.040","Text":"b is simply the matrix of the 4 partial derivatives of F with respect to x,"},{"Start":"05:28.040 ","End":"05:30.080","Text":"F with respect to y, G with respect to x,"},{"Start":"05:30.080 ","End":"05:33.770","Text":"G with respect to y at that particular point, a, b."},{"Start":"05:33.770 ","End":"05:37.470","Text":"Again, in the example, it will be clearer."},{"Start":"05:38.050 ","End":"05:41.870","Text":"Then we do this for each critical point a, b."},{"Start":"05:41.870 ","End":"05:43.760","Text":"For each critical point,"},{"Start":"05:43.760 ","End":"05:45.805","Text":"we have a matrix,"},{"Start":"05:45.805 ","End":"05:48.255","Text":"2 by 2 constants."},{"Start":"05:48.255 ","End":"05:50.570","Text":"Then, as before, we classify it."},{"Start":"05:50.570 ","End":"05:57.555","Text":"It might be a saddle and unstable, and so on."},{"Start":"05:57.555 ","End":"06:01.850","Text":"It turns out this is not 100 percent accurate because of the approximation."},{"Start":"06:01.850 ","End":"06:06.390","Text":"Sometimes it misses us to what the actual type is."},{"Start":"06:06.560 ","End":"06:16.580","Text":"I\u0027d like to point out, that this matrix A is simply the value of the Jacobian matrix."},{"Start":"06:16.580 ","End":"06:20.150","Text":"Remember the partial derivatives of F and G,"},{"Start":"06:20.150 ","End":"06:22.235","Text":"but these are functions of x and y."},{"Start":"06:22.235 ","End":"06:26.540","Text":"We evaluate at each critical point that x=a,"},{"Start":"06:26.540 ","End":"06:29.585","Text":"y=b, and then we get a matrix of constants."},{"Start":"06:29.585 ","End":"06:31.100","Text":"You might get 3 of these,"},{"Start":"06:31.100 ","End":"06:33.470","Text":"As if there are 3 critical points."},{"Start":"06:33.470 ","End":"06:35.615","Text":"They\u0027re all just the Jacobian,"},{"Start":"06:35.615 ","End":"06:38.045","Text":"at a particular critical point."},{"Start":"06:38.045 ","End":"06:44.259","Text":"I\u0027ll just say something general about the sketching of such a system phase portrait,"},{"Start":"06:44.259 ","End":"06:48.870","Text":"and then we\u0027ll get to the example and be less abstract."},{"Start":"06:48.870 ","End":"06:51.035","Text":"We get these critical points."},{"Start":"06:51.035 ","End":"06:52.340","Text":"I say, there might not be any,"},{"Start":"06:52.340 ","End":"06:55.144","Text":"there might be 1, there might be several."},{"Start":"06:55.144 ","End":"06:57.940","Text":"We classify each one."},{"Start":"06:57.940 ","End":"07:01.700","Text":"We might say, you have a saddle over here,"},{"Start":"07:01.700 ","End":"07:08.840","Text":"and asymptotically stable star point here, and so on."},{"Start":"07:08.840 ","End":"07:12.900","Text":"That helps us to get a general idea."},{"Start":"07:12.900 ","End":"07:20.000","Text":"We could give a rough sketch of each critical point, each classification."},{"Start":"07:20.000 ","End":"07:26.839","Text":"Then somehow, add extra lines and join it into the picture and get a very rough idea."},{"Start":"07:26.839 ","End":"07:28.790","Text":"It doesn\u0027t give us very much to work this way."},{"Start":"07:28.790 ","End":"07:31.890","Text":"You really have to work computerized."},{"Start":"07:32.180 ","End":"07:36.205","Text":"Like I said, also, on the Internet,"},{"Start":"07:36.205 ","End":"07:40.910","Text":"there are a lot of sites that do this kind of calculation for you."},{"Start":"07:40.910 ","End":"07:45.170","Text":"I\u0027m not going to give any because they go out of date so quickly."},{"Start":"07:47.180 ","End":"07:52.035","Text":"Onto our first example."},{"Start":"07:52.035 ","End":"07:55.275","Text":"I went on, and I give myself some more room,"},{"Start":"07:55.275 ","End":"08:01.045","Text":"and x\u0027 is this and y\u0027 is this."},{"Start":"08:01.045 ","End":"08:03.520","Text":"Each of them is a function of x and y,"},{"Start":"08:03.520 ","End":"08:06.910","Text":"and no ts appear in these."},{"Start":"08:06.910 ","End":"08:14.085","Text":"It\u0027s one of our autonomous planar non-linear systems."},{"Start":"08:14.085 ","End":"08:20.290","Text":"Like I said, we find the critical points by setting x\u0027 and y\u0027 each of them to 0,"},{"Start":"08:20.290 ","End":"08:23.230","Text":"which gives us 2x minus y is 0,"},{"Start":"08:23.230 ","End":"08:26.695","Text":"x^2 plus y^2 minus 5 is 0."},{"Start":"08:26.695 ","End":"08:29.545","Text":"We can solve this."},{"Start":"08:29.545 ","End":"08:32.110","Text":"There are 2 solutions, and here they are."},{"Start":"08:32.110 ","End":"08:33.835","Text":"I\u0027ll just show you how you might do it."},{"Start":"08:33.835 ","End":"08:36.655","Text":"From here, you would plug in y=2x,"},{"Start":"08:36.655 ","End":"08:39.685","Text":"and you plug 2x instead of y here,"},{"Start":"08:39.685 ","End":"08:43.600","Text":"so you get x^2 plus 4x^2 minus 5 is 0."},{"Start":"08:43.600 ","End":"08:45.400","Text":"You get x^2 equals 1."},{"Start":"08:45.400 ","End":"08:47.785","Text":"X is plus or minus 1."},{"Start":"08:47.785 ","End":"08:49.895","Text":"Since y is 2x,"},{"Start":"08:49.895 ","End":"08:53.055","Text":"1 gives us 2 and minus 1 gives us minus 2."},{"Start":"08:53.055 ","End":"08:54.480","Text":"There\u0027s 2 solutions,"},{"Start":"08:54.480 ","End":"08:57.165","Text":"and so 2 critical points."},{"Start":"08:57.165 ","End":"09:02.040","Text":"Next, we\u0027ll need the Jacobian for these 2 functions,"},{"Start":"09:02.040 ","End":"09:05.380","Text":"F and G. In general,"},{"Start":"09:05.380 ","End":"09:09.489","Text":"it\u0027s the partial derivatives of each of them with respect to x and y."},{"Start":"09:09.489 ","End":"09:11.140","Text":"If we look here,"},{"Start":"09:11.140 ","End":"09:12.795","Text":"at what F and G are,"},{"Start":"09:12.795 ","End":"09:15.760","Text":"F with respect to x is 2x minus y."},{"Start":"09:15.760 ","End":"09:17.404","Text":"Derivative with respect to x is 2,"},{"Start":"09:17.404 ","End":"09:19.412","Text":"similarly for the others."},{"Start":"09:19.412 ","End":"09:20.665","Text":"This is what we get."},{"Start":"09:20.665 ","End":"09:23.185","Text":"But we have to evaluate this Jacobian"},{"Start":"09:23.185 ","End":"09:26.960","Text":"at each of the critical points, and there\u0027s 2 of them."},{"Start":"09:26.960 ","End":"09:29.580","Text":"If we plug in x=1,"},{"Start":"09:29.580 ","End":"09:33.695","Text":"y=2 here, this is what we get."},{"Start":"09:33.695 ","End":"09:36.130","Text":"If we plug in the other one,"},{"Start":"09:36.130 ","End":"09:39.330","Text":"where x is minus 1 and y is minus 2,"},{"Start":"09:39.330 ","End":"09:42.100","Text":"then this is what we get."},{"Start":"09:42.680 ","End":"09:47.090","Text":"For this matrix, the eigenvalues,"},{"Start":"09:47.090 ","End":"09:48.275","Text":"if you compute them,"},{"Start":"09:48.275 ","End":"09:53.600","Text":"come out to be a complex conjugate pair 3 plus or minus i."},{"Start":"09:53.600 ","End":"09:55.520","Text":"Whenever it\u0027s a real matrix,"},{"Start":"09:55.520 ","End":"09:57.290","Text":"if you get complex,"},{"Start":"09:57.290 ","End":"09:59.330","Text":"then they\u0027re conjugate pairs."},{"Start":"09:59.330 ","End":"10:02.330","Text":"The real part is 3, which is positive."},{"Start":"10:02.330 ","End":"10:03.950","Text":"If you go to that table,"},{"Start":"10:03.950 ","End":"10:13.210","Text":"complex conjugate pair with positive real part is a spiral point, and it\u0027s unstable."},{"Start":"10:13.310 ","End":"10:15.920","Text":"If the real part was negative,"},{"Start":"10:15.920 ","End":"10:18.515","Text":"it would be asymptotically stable."},{"Start":"10:18.515 ","End":"10:21.630","Text":"As for the other one,"},{"Start":"10:21.630 ","End":"10:25.490","Text":"this time, they\u0027re both real,"},{"Start":"10:25.490 ","End":"10:27.695","Text":"but they have opposite signs."},{"Start":"10:27.695 ","End":"10:31.580","Text":"Obviously, root 11 is bigger than 1."},{"Start":"10:31.580 ","End":"10:34.790","Text":"It\u0027s even bigger than 3 because it\u0027s bigger than root 9,"},{"Start":"10:34.790 ","End":"10:37.520","Text":"so minus 1 plus this is positive."},{"Start":"10:37.520 ","End":"10:39.620","Text":"Minus 1 minus this is negative,"},{"Start":"10:39.620 ","End":"10:42.110","Text":"so we have the case real with opposite signs."},{"Start":"10:42.110 ","End":"10:45.470","Text":"You go up to the table of cases."},{"Start":"10:45.470 ","End":"10:48.170","Text":"When this happens, it\u0027s always a saddle point,"},{"Start":"10:48.170 ","End":"10:50.345","Text":"which is always unstable."},{"Start":"10:50.345 ","End":"10:52.775","Text":"Now, we know that we have 2 critical points."},{"Start":"10:52.775 ","End":"10:57.480","Text":"One is the spiral,"},{"Start":"10:57.480 ","End":"10:59.310","Text":"and one is the saddle."},{"Start":"10:59.310 ","End":"11:03.770","Text":"That could give us a rough idea for starting to sketch,"},{"Start":"11:03.770 ","End":"11:07.385","Text":"but we\u0027ll go to the computer-aided."},{"Start":"11:07.385 ","End":"11:10.925","Text":"If it\u0027s a computer program,"},{"Start":"11:10.925 ","End":"11:12.620","Text":"you feed it various parameters."},{"Start":"11:12.620 ","End":"11:16.790","Text":"For example, I tell it that I want to go from minus 5 to 5."},{"Start":"11:16.790 ","End":"11:21.200","Text":"You can tell it if you want the arrows to be scaled or not,"},{"Start":"11:21.200 ","End":"11:22.790","Text":"or they\u0027re all the same length,"},{"Start":"11:22.790 ","End":"11:26.255","Text":"and how densely you want them, and so on."},{"Start":"11:26.255 ","End":"11:30.925","Text":"Anyway, this is the one way of getting a sketch."},{"Start":"11:30.925 ","End":"11:38.745","Text":"In this program computer calculator for this thing,"},{"Start":"11:38.745 ","End":"11:40.880","Text":"first of all, it gives you the direction field."},{"Start":"11:40.880 ","End":"11:43.400","Text":"Then you touch it in various places,"},{"Start":"11:43.400 ","End":"11:46.805","Text":"and it draws the solution trajectory orbit through that."},{"Start":"11:46.805 ","End":"11:48.215","Text":"I\u0027ve done a few of those,"},{"Start":"11:48.215 ","End":"11:53.150","Text":"and we end up with something like this,"},{"Start":"11:53.150 ","End":"11:54.950","Text":"which shows us here,"},{"Start":"11:54.950 ","End":"12:01.055","Text":"we have the 0.12 with the spiral point."},{"Start":"12:01.055 ","End":"12:03.790","Text":"I still have to put the arrows in."},{"Start":"12:03.790 ","End":"12:06.620","Text":"Well, the arrows are going to align with these arrows."},{"Start":"12:06.620 ","End":"12:09.410","Text":"This one\u0027s going this way."},{"Start":"12:09.410 ","End":"12:11.915","Text":"At least, one arrow on each of them."},{"Start":"12:11.915 ","End":"12:16.220","Text":"I don\u0027t know how much I have patience for, but anyway,"},{"Start":"12:16.220 ","End":"12:19.160","Text":"it shouldn\u0027t take too long here, and here,"},{"Start":"12:19.160 ","End":"12:24.465","Text":"and so on here."},{"Start":"12:24.465 ","End":"12:29.784","Text":"I\u0027ll stop now. You see that here,"},{"Start":"12:29.784 ","End":"12:32.710","Text":"they\u0027re all going away from the point."},{"Start":"12:32.710 ","End":"12:36.870","Text":"This is obviously the critical point at 1,"},{"Start":"12:36.870 ","End":"12:39.285","Text":"2, and minus 1,"},{"Start":"12:39.285 ","End":"12:42.030","Text":"minus 2 is here."},{"Start":"12:42.030 ","End":"12:46.449","Text":"It looks like the saddle of a couple of crosshairs,"},{"Start":"12:46.449 ","End":"12:47.630","Text":"like, straight lines through."},{"Start":"12:47.630 ","End":"12:50.560","Text":"Actually, this is a solution all into itself."},{"Start":"12:50.560 ","End":"12:53.515","Text":"Mostly, they\u0027re all going away from the point, from the saddle."},{"Start":"12:53.515 ","End":"12:56.668","Text":"It\u0027s always unstable. Anyway,"},{"Start":"12:56.668 ","End":"12:58.450","Text":"this is the general idea."},{"Start":"12:58.450 ","End":"13:03.249","Text":"This is the sketch phase portrait,"},{"Start":"13:03.249 ","End":"13:08.025","Text":"meaning, we have a few representative solutions. That\u0027s it."},{"Start":"13:08.025 ","End":"13:10.705","Text":"There will be plenty, I hope,"},{"Start":"13:10.705 ","End":"13:14.750","Text":"of solved examples following the tutorial,"},{"Start":"13:14.750 ","End":"13:19.205","Text":"and you\u0027ll learn a lot through those solved examples."},{"Start":"13:19.205 ","End":"13:21.540","Text":"I\u0027m done for now."}],"ID":10708},{"Watched":false,"Name":"exercise 1","Duration":"8m 20s","ChapterTopicVideoID":10359,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.970","Text":"In this exercise, we have a system of"},{"Start":"00:02.970 ","End":"00:08.100","Text":"differential equations in the plane written in matrix form."},{"Start":"00:08.100 ","End":"00:11.850","Text":"We have x, which sometimes we write x,"},{"Start":"00:11.850 ","End":"00:13.350","Text":"y sometimes the x_1,"},{"Start":"00:13.350 ","End":"00:17.990","Text":"x_2 and A is a constant matrix."},{"Start":"00:17.990 ","End":"00:20.225","Text":"We have the equation,"},{"Start":"00:20.225 ","End":"00:25.249","Text":"which really is a system of equations, x\u0027 equals Ax."},{"Start":"00:25.249 ","End":"00:29.570","Text":"In this case, we take it from physics that it\u0027s the position of"},{"Start":"00:29.570 ","End":"00:34.520","Text":"a particle in the plane at time t. We have 2 parts."},{"Start":"00:34.520 ","End":"00:41.705","Text":"First, find the general solution for x of t of this differential equation and secondly,"},{"Start":"00:41.705 ","End":"00:44.675","Text":"we have to classify the critical points."},{"Start":"00:44.675 ","End":"00:50.330","Text":"What kind is it and how stable is it and to sketch"},{"Start":"00:50.330 ","End":"00:57.480","Text":"rough sketch of a phase portrait or phase plane portrait of the system in the plane."},{"Start":"00:58.100 ","End":"01:02.230","Text":"I\u0027m going to gloss over part a pretty quickly because I really want to"},{"Start":"01:02.230 ","End":"01:06.625","Text":"concentrate on the phase plane portrait part."},{"Start":"01:06.625 ","End":"01:09.130","Text":"I\u0027ll just give you the general steps."},{"Start":"01:09.130 ","End":"01:12.670","Text":"First of all, you\u0027ll find the eigenvalues of the matrix A,"},{"Start":"01:12.670 ","End":"01:14.120","Text":"which you know how to do."},{"Start":"01:14.120 ","End":"01:21.560","Text":"The answer comes out to be 2 minus 1 and minus 5 column Lambda 1, Lambda 2."},{"Start":"01:21.560 ","End":"01:27.095","Text":"Next we find the corresponding eigenvectors of A."},{"Start":"01:27.095 ","End":"01:30.185","Text":"For the first Lambda minus 1,"},{"Start":"01:30.185 ","End":"01:35.200","Text":"we get this as an eigenvector and for minus 5, we get this."},{"Start":"01:35.200 ","End":"01:40.895","Text":"Of course you could get a non-zero multiple of 1 of these unique."},{"Start":"01:40.895 ","End":"01:44.780","Text":"Next we get a fundamental set of solutions,"},{"Start":"01:44.780 ","End":"01:52.560","Text":"which in this case we just take e to the power of this Lambda times t times this vector."},{"Start":"01:52.560 ","End":"01:58.715","Text":"The other 1 is e to the power of this times t times this vector."},{"Start":"01:58.715 ","End":"02:05.815","Text":"Then we combine these 2 to get the general solution,"},{"Start":"02:05.815 ","End":"02:10.760","Text":"which is just linear combination of these 2,"},{"Start":"02:10.760 ","End":"02:18.730","Text":"say c_1 times this solution plus c_2 times the other solution."},{"Start":"02:18.730 ","End":"02:22.300","Text":"Moving on to part B,"},{"Start":"02:22.300 ","End":"02:29.450","Text":"the critical points are found by letting Ax equals 0."},{"Start":"02:29.450 ","End":"02:34.220","Text":"But the matrix a is non-singular,"},{"Start":"02:34.220 ","End":"02:36.305","Text":"meaning the determinant is not 0."},{"Start":"02:36.305 ","End":"02:43.610","Text":"Whenever you get 2 distinct eigenvalues or n distinct eigenvalues and n dimensions,"},{"Start":"02:43.610 ","End":"02:49.595","Text":"then we\u0027re non-singular and we have only the origin as a critical point."},{"Start":"02:49.595 ","End":"02:52.130","Text":"Now we just want to classify it."},{"Start":"02:52.130 ","End":"03:00.660","Text":"In fact, there is a table that helps us to classify according to the eigenvalues."},{"Start":"03:00.660 ","End":"03:04.970","Text":"Notice that we have 2 distinct negative eigenvalues."},{"Start":"03:04.970 ","End":"03:09.380","Text":"When this happens, then the type of critical point"},{"Start":"03:09.380 ","End":"03:15.060","Text":"is a node and it\u0027s also asymptotically stable."},{"Start":"03:16.210 ","End":"03:21.125","Text":"Now we just need to do the sketch."},{"Start":"03:21.125 ","End":"03:23.690","Text":"We\u0027ll do this on a new page."},{"Start":"03:23.690 ","End":"03:26.405","Text":"I dragged the result with me."},{"Start":"03:26.405 ","End":"03:28.880","Text":"Start with a pair of axes."},{"Start":"03:28.880 ","End":"03:30.230","Text":"Remember here when NOT x, y,"},{"Start":"03:30.230 ","End":"03:32.045","Text":"we\u0027re x_1 and x_2."},{"Start":"03:32.045 ","End":"03:39.400","Text":"Then you need to draw lines through the origin parallel to these 2 eigenvectors."},{"Start":"03:39.400 ","End":"03:42.360","Text":"Let me explain what I did here."},{"Start":"03:42.360 ","End":"03:46.605","Text":"You would start by plotting the 2 eigenvectors."},{"Start":"03:46.605 ","End":"03:50.480","Text":"Like 1, 3 would be the vector from here to,"},{"Start":"03:50.480 ","End":"03:54.765","Text":"say this is 1 and this is 3 somewhere here 1,"},{"Start":"03:54.765 ","End":"03:58.035","Text":"3 and then the other 1,"},{"Start":"03:58.035 ","End":"04:01.725","Text":"minus 1, 1 maybe here,"},{"Start":"04:01.725 ","End":"04:03.855","Text":"minus 1, 1."},{"Start":"04:03.855 ","End":"04:06.210","Text":"Then we draw the lines through those."},{"Start":"04:06.210 ","End":"04:11.580","Text":"I\u0027m going to explain about the arrows and how that works."},{"Start":"04:11.810 ","End":"04:14.390","Text":"I\u0027m going to break this thing up into cases."},{"Start":"04:14.390 ","End":"04:16.820","Text":"Let\u0027s say first of all, if c_2 is 0,"},{"Start":"04:16.820 ","End":"04:18.755","Text":"then we just have this part."},{"Start":"04:18.755 ","End":"04:23.505","Text":"Now, if c_1 is positive,"},{"Start":"04:23.505 ","End":"04:27.375","Text":"then e to the minus t is also positive."},{"Start":"04:27.375 ","End":"04:30.885","Text":"We get positive multiples of 1, 3."},{"Start":"04:30.885 ","End":"04:35.315","Text":"We get the part of the line that\u0027s in the first quadrant."},{"Start":"04:35.315 ","End":"04:38.780","Text":"If c_1 is negative,"},{"Start":"04:38.780 ","End":"04:41.990","Text":"then we\u0027re going to have this path of the line."},{"Start":"04:41.990 ","End":"04:45.560","Text":"Notice I put the arrows this way because as t"},{"Start":"04:45.560 ","End":"04:50.165","Text":"goes to infinity, e to the minus t goes to 0."},{"Start":"04:50.165 ","End":"04:51.830","Text":"When t goes to infinity,"},{"Start":"04:51.830 ","End":"04:54.830","Text":"in either case we\u0027re moving towards the origin,"},{"Start":"04:54.830 ","End":"04:58.950","Text":"which is the critical point."},{"Start":"04:59.050 ","End":"05:05.720","Text":"Then similarly, we\u0027d get a similar thing if c_1 is 0."},{"Start":"05:05.720 ","End":"05:08.555","Text":"Then if c_2 is positive,"},{"Start":"05:08.555 ","End":"05:10.385","Text":"then we get this part."},{"Start":"05:10.385 ","End":"05:11.750","Text":"If c_2 is negative,"},{"Start":"05:11.750 ","End":"05:12.890","Text":"we get this part."},{"Start":"05:12.890 ","End":"05:15.739","Text":"In this case also when t goes to infinity,"},{"Start":"05:15.739 ","End":"05:18.335","Text":"we\u0027re moving towards the origin."},{"Start":"05:18.335 ","End":"05:20.725","Text":"Let me just write that."},{"Start":"05:20.725 ","End":"05:22.985","Text":"Here I put it in words."},{"Start":"05:22.985 ","End":"05:31.620","Text":"Also, there\u0027s the formulas for the lines that this line here is where x_2 is 3x_1,"},{"Start":"05:31.620 ","End":"05:37.625","Text":"because you can see it\u0027s the x_2 part is 3 times the x_1 part."},{"Start":"05:37.625 ","End":"05:42.560","Text":"Here we get x_2 is minus x_1."},{"Start":"05:42.560 ","End":"05:48.510","Text":"Now we need the third case where neither of these is 0."},{"Start":"05:48.920 ","End":"05:53.380","Text":"Then we want to study what happens at"},{"Start":"05:53.380 ","End":"05:59.800","Text":"the trajectories asymptotically meaning when t goes to infinity or minus infinity."},{"Start":"05:59.800 ","End":"06:04.085","Text":"I\u0027m going to switch the picture first and then I\u0027ll explain."},{"Start":"06:04.085 ","End":"06:06.510","Text":"Here\u0027s the sketch."},{"Start":"06:06.510 ","End":"06:10.240","Text":"We had the blue lines before,"},{"Start":"06:10.240 ","End":"06:16.915","Text":"but now I added these black trajectories and I\u0027m going to explain below."},{"Start":"06:16.915 ","End":"06:20.980","Text":"Basically, I\u0027m going to show you that they approach 0 as t"},{"Start":"06:20.980 ","End":"06:25.480","Text":"goes to infinity asymptotically along this line from here,"},{"Start":"06:25.480 ","End":"06:29.635","Text":"from here, or along this part of the line from here and from here."},{"Start":"06:29.635 ","End":"06:31.730","Text":"But they also come from,"},{"Start":"06:31.730 ","End":"06:35.495","Text":"if you go back to minus infinity for t,"},{"Start":"06:35.495 ","End":"06:39.710","Text":"they come from a line that\u0027s parallel to this line."},{"Start":"06:39.710 ","End":"06:41.045","Text":"This is not an asymptote,"},{"Start":"06:41.045 ","End":"06:45.395","Text":"but they come from a parallel asymptote."},{"Start":"06:45.395 ","End":"06:52.265","Text":"Let me explain all that."},{"Start":"06:52.265 ","End":"06:55.430","Text":"When t goes to infinity,"},{"Start":"06:55.430 ","End":"06:58.805","Text":"the trajectories tend towards the origin."},{"Start":"06:58.805 ","End":"07:02.850","Text":"We\u0027ve lost the formula,"},{"Start":"07:02.850 ","End":"07:03.930","Text":"but if you look back,"},{"Start":"07:03.930 ","End":"07:07.920","Text":"you\u0027ll see that 1 of them has an e to the minus t and 1 of them has"},{"Start":"07:07.920 ","End":"07:14.580","Text":"an e to the minus 5t."},{"Start":"07:14.580 ","End":"07:17.300","Text":"This is much larger than this."},{"Start":"07:17.300 ","End":"07:21.020","Text":"This is the term that dominates this."},{"Start":"07:21.020 ","End":"07:26.870","Text":"The trajectories go to x_1."},{"Start":"07:26.870 ","End":"07:28.670","Text":"If I scroll back to the formula,"},{"Start":"07:28.670 ","End":"07:32.510","Text":"you\u0027ll see that just the c_1 part counts,"},{"Start":"07:32.510 ","End":"07:34.630","Text":"the c_2 is negligible."},{"Start":"07:34.630 ","End":"07:36.440","Text":"On the other hand,"},{"Start":"07:36.440 ","End":"07:39.515","Text":"when t goes to minus infinity,"},{"Start":"07:39.515 ","End":"07:45.290","Text":"the trajectories move away from the origin because then t goes to minus infinity,"},{"Start":"07:45.290 ","End":"07:51.200","Text":"minus t goes to infinity and then the e^5t part is on a much older magnitude"},{"Start":"07:51.200 ","End":"08:00.000","Text":"than e to the minus t. Really the c_2 part counts more."},{"Start":"08:00.290 ","End":"08:04.081","Text":"If we look at the formula,"},{"Start":"08:04.081 ","End":"08:07.400","Text":"this part is the more important part, but still,"},{"Start":"08:07.400 ","End":"08:11.270","Text":"this is not totally negligible so that these things"},{"Start":"08:11.270 ","End":"08:15.980","Text":"go not towards this as an asymptote, but only parallel."},{"Start":"08:15.980 ","End":"08:21.150","Text":"That\u0027s basically it. We\u0027re done."}],"ID":10709},{"Watched":false,"Name":"exercise 2","Duration":"8m 41s","ChapterTopicVideoID":10345,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.180","Text":"In this exercise, we have a system of differential equations; x\u0027 = Ax."},{"Start":"00:06.180 ","End":"00:07.335","Text":"It\u0027s in matrix form."},{"Start":"00:07.335 ","End":"00:10.530","Text":"A is a constant matrix, this one,"},{"Start":"00:10.530 ","End":"00:12.750","Text":"and x is a vector,"},{"Start":"00:12.750 ","End":"00:14.970","Text":"a 2D vector, x_1, x_2,"},{"Start":"00:14.970 ","End":"00:18.960","Text":"everything is a function of t. In this case from physics,"},{"Start":"00:18.960 ","End":"00:24.300","Text":"t is time and x is the position of a particle in"},{"Start":"00:24.300 ","End":"00:29.910","Text":"the plane at time t. We have 2 tasks."},{"Start":"00:29.910 ","End":"00:35.385","Text":"One is to find the general solution x as a function of t for this differential equation,"},{"Start":"00:35.385 ","End":"00:37.440","Text":"actually for this system of"},{"Start":"00:37.440 ","End":"00:42.350","Text":"differential equations and we have to classify the critical point."},{"Start":"00:42.350 ","End":"00:46.640","Text":"We know there\u0027s only one. This kind of system only has the origin as a critical point."},{"Start":"00:46.640 ","End":"00:50.408","Text":"We have to classify it and that means"},{"Start":"00:50.408 ","End":"00:55.415","Text":"whether it\u0027s a node or a saddle or whatever and if it\u0027s stable or unstable,"},{"Start":"00:55.415 ","End":"01:00.480","Text":"and to sketch a phase portrait of the system in the plane."},{"Start":"01:01.730 ","End":"01:07.275","Text":"Part A is more routine and should be familiar so we just whizzed through that one."},{"Start":"01:07.275 ","End":"01:12.350","Text":"B Is the more important one and we\u0027re learning more about phase portraits."},{"Start":"01:12.350 ","End":"01:13.910","Text":"So we\u0027ll just go through it quickly."},{"Start":"01:13.910 ","End":"01:18.230","Text":"What we do is we take the matrix and we find the eigenvalues and"},{"Start":"01:18.230 ","End":"01:23.730","Text":"I\u0027m just giving you the answers that came out to be minus 2 and 3."},{"Start":"01:23.890 ","End":"01:26.450","Text":"After we find the eigenvalues,"},{"Start":"01:26.450 ","End":"01:29.405","Text":"we find the corresponding eigenvectors."},{"Start":"01:29.405 ","End":"01:31.310","Text":"This one corresponds to this one,"},{"Start":"01:31.310 ","End":"01:34.050","Text":"this one corresponds to this one."},{"Start":"01:34.480 ","End":"01:39.950","Text":"Next, we build a set of fundamental solutions,"},{"Start":"01:39.950 ","End":"01:42.380","Text":"is only going to be 2 because it\u0027s in 2D."},{"Start":"01:42.380 ","End":"01:46.820","Text":"What we do is it\u0027s e to the power of this eigenvalue,"},{"Start":"01:46.820 ","End":"01:54.275","Text":"t times this eigenvector and similarly for the other eigenvalue and eigenvector."},{"Start":"01:54.275 ","End":"02:01.820","Text":"From this we want to go to a general solution which is just a linear combination."},{"Start":"02:01.820 ","End":"02:05.765","Text":"So that was Part A. Let\u0027s move on to Part B."},{"Start":"02:05.765 ","End":"02:14.340","Text":"We get the critical points by solving the differential equation where x\u0027=0,"},{"Start":"02:14.410 ","End":"02:21.770","Text":"x\u0027 is Ax and A is the 2 by 2 matrix while it\u0027s scrolled off."},{"Start":"02:21.770 ","End":"02:24.214","Text":"But if we go back and check,"},{"Start":"02:24.214 ","End":"02:28.580","Text":"you\u0027ll see that it\u0027s a non-singular matrix."},{"Start":"02:28.580 ","End":"02:30.575","Text":"The determinant is not 0,"},{"Start":"02:30.575 ","End":"02:35.705","Text":"so the only solution is 0,0, the origin."},{"Start":"02:35.705 ","End":"02:41.105","Text":"The classification we get by looking at the eigenvalues,"},{"Start":"02:41.105 ","End":"02:43.750","Text":"we have a negative and a positive eigenvalue."},{"Start":"02:43.750 ","End":"02:51.590","Text":"Whenever this happens, then it\u0027s always a saddle point and it\u0027s unstable."},{"Start":"02:51.590 ","End":"02:59.239","Text":"All that remains now is the sketch parts and we\u0027ll do that on a fresh page."},{"Start":"02:59.239 ","End":"03:04.585","Text":"Here we are and I just brought along the general solution as a reminder."},{"Start":"03:04.585 ","End":"03:06.815","Text":"We start out with a pair of axes,"},{"Start":"03:06.815 ","End":"03:11.900","Text":"and on these axis we draw the lines that correspond to the eigenvectors."},{"Start":"03:11.900 ","End":"03:18.524","Text":"Like this here might be the point minus 2,"},{"Start":"03:18.524 ","End":"03:25.755","Text":"3, and here might be the point 1,1 and we just draw the lines through these."},{"Start":"03:25.755 ","End":"03:28.790","Text":"These lines actually correspond to 2 special cases."},{"Start":"03:28.790 ","End":"03:32.705","Text":"One where c_2 is 0 and the other was c_1 is 0."},{"Start":"03:32.705 ","End":"03:35.614","Text":"Let\u0027s start with the one where c_2 is 0."},{"Start":"03:35.614 ","End":"03:37.040","Text":"And if c_2 is 0,"},{"Start":"03:37.040 ","End":"03:43.015","Text":"we subdivide into 2 cases where c_1 is positive and c_1 is negative."},{"Start":"03:43.015 ","End":"03:44.960","Text":"Actually to be precise,"},{"Start":"03:44.960 ","End":"03:52.940","Text":"we should cut out a hole where the origin is because the origin is a solution on its own,"},{"Start":"03:52.940 ","End":"03:54.770","Text":"it\u0027s the equilibrium solution."},{"Start":"03:54.770 ","End":"03:56.750","Text":"We said that c_2 is 0."},{"Start":"03:56.750 ","End":"03:58.595","Text":"If c_1 is positive,"},{"Start":"03:58.595 ","End":"04:08.390","Text":"this thing is also positive so we get this part up to the origin in the second quadrant."},{"Start":"04:08.390 ","End":"04:11.120","Text":"When c_1 is negative,"},{"Start":"04:11.120 ","End":"04:15.497","Text":"we get the other half of the line in the fourth quadrant."},{"Start":"04:15.497 ","End":"04:17.480","Text":"Similarly, if I do the opposite,"},{"Start":"04:17.480 ","End":"04:20.840","Text":"if I let c_1=0,"},{"Start":"04:20.840 ","End":"04:22.880","Text":"then when c_2 is positive,"},{"Start":"04:22.880 ","End":"04:28.710","Text":"I get the part in the first quadrant and when c_2 is negative,"},{"Start":"04:28.710 ","End":"04:31.205","Text":"I get the parts in the third quadrant."},{"Start":"04:31.205 ","End":"04:37.940","Text":"Notice that when t goes towards infinity,"},{"Start":"04:37.940 ","End":"04:42.620","Text":"then for this line because of the e to the minus 3t,"},{"Start":"04:42.620 ","End":"04:47.300","Text":"then everything goes towards the origin in either case,"},{"Start":"04:47.300 ","End":"04:53.785","Text":"whether c_1 is bigger than 0 or less than 0."},{"Start":"04:53.785 ","End":"04:55.835","Text":"In the case of the other line,"},{"Start":"04:55.835 ","End":"05:03.790","Text":"when we have the e^2t,"},{"Start":"05:03.790 ","End":"05:08.720","Text":"that goes towards infinity."},{"Start":"05:10.080 ","End":"05:18.475","Text":"In either case, just go back and look at that, yeah, see there\u0027s e to the minus 2t,"},{"Start":"05:18.475 ","End":"05:25.260","Text":"which goes to 0 when t goes to infinity,"},{"Start":"05:25.260 ","End":"05:28.205","Text":"and e to the 3t goes to infinity,"},{"Start":"05:28.205 ","End":"05:29.770","Text":"when t goes to infinity."},{"Start":"05:29.770 ","End":"05:33.130","Text":"So here on this line we\u0027re going away from"},{"Start":"05:33.130 ","End":"05:37.640","Text":"the origin towards infinity and here we\u0027re coming towards the origin."},{"Start":"05:37.760 ","End":"05:40.755","Text":"Just put a few extra arrows,"},{"Start":"05:40.755 ","End":"05:42.800","Text":"looks a bit nicer now."},{"Start":"05:42.800 ","End":"05:51.215","Text":"So far we\u0027ve covered the cases where c_1 is 0 or c_2 is 0 and this is the picture we get."},{"Start":"05:51.215 ","End":"05:53.600","Text":"Just to remind you,"},{"Start":"05:53.600 ","End":"05:55.234","Text":"this was the formula."},{"Start":"05:55.234 ","End":"06:00.965","Text":"Now we\u0027re going to take the case where both are not 0, neither of these."},{"Start":"06:00.965 ","End":"06:07.640","Text":"What we really care about is what happens when t goes to infinity or minus infinity."},{"Start":"06:07.640 ","End":"06:09.995","Text":"Look, if t goes to infinity,"},{"Start":"06:09.995 ","End":"06:12.260","Text":"this bit goes to infinity,"},{"Start":"06:12.260 ","End":"06:17.030","Text":"and this becomes negligible almost 0 and the other way round,"},{"Start":"06:17.030 ","End":"06:18.980","Text":"if t goes to minus infinity,"},{"Start":"06:18.980 ","End":"06:22.124","Text":"this e to the minus 2t goes to infinity"},{"Start":"06:22.124 ","End":"06:26.370","Text":"so we\u0027re going to get far away and this will be negligible."},{"Start":"06:27.730 ","End":"06:30.350","Text":"We\u0027re going to take both of them,"},{"Start":"06:30.350 ","End":"06:34.795","Text":"both cancels non-zero and see what happens asymptotically."},{"Start":"06:34.795 ","End":"06:38.005","Text":"The first case was when t goes to infinity,"},{"Start":"06:38.005 ","End":"06:41.290","Text":"then the trajectories are going to move away from the origin."},{"Start":"06:41.290 ","End":"06:46.750","Text":"But more than that, it\u0027s going to be in the direction of the second vector,"},{"Start":"06:46.750 ","End":"06:51.595","Text":"let\u0027s see what it was, this one."},{"Start":"06:51.595 ","End":"06:59.110","Text":"So asymptotically, it\u0027s going to go to this line here."},{"Start":"06:59.110 ","End":"07:03.595","Text":"When we go to minus infinity,"},{"Start":"07:03.595 ","End":"07:06.955","Text":"we\u0027re going to get close to this line here."},{"Start":"07:06.955 ","End":"07:10.010","Text":"Let me just give you an example."},{"Start":"07:10.470 ","End":"07:18.440","Text":"At infinity, might go this way and at minus infinity towards here."},{"Start":"07:18.510 ","End":"07:23.925","Text":"There\u0027s various ones like these."},{"Start":"07:23.925 ","End":"07:28.820","Text":"Similarly, here we\u0027re going to go from minus infinity to infinity,"},{"Start":"07:28.820 ","End":"07:36.710","Text":"from this asymptote to this asymptote and now let me replace this with a proper sketch."},{"Start":"07:36.710 ","End":"07:41.955","Text":"Here we are. So it would look something like this."},{"Start":"07:41.955 ","End":"07:44.180","Text":"Just a rough idea."},{"Start":"07:44.180 ","End":"07:47.120","Text":"You don\u0027t have to do it even this accurately."},{"Start":"07:47.120 ","End":"07:50.840","Text":"A proper way to do it is maybe with a computer aided,"},{"Start":"07:50.840 ","End":"07:52.925","Text":"but we just want to get the general idea."},{"Start":"07:52.925 ","End":"07:55.850","Text":"I think this will do,"},{"Start":"07:55.850 ","End":"07:59.110","Text":"this one I write that bit about the asymptotes."},{"Start":"07:59.110 ","End":"08:02.660","Text":"This part is negligible."},{"Start":"08:02.660 ","End":"08:09.635","Text":"This part dominates and we go towards the line x2=x1, which is this one."},{"Start":"08:09.635 ","End":"08:12.695","Text":"When t goes to minus infinity,"},{"Start":"08:12.695 ","End":"08:18.080","Text":"trajectories move away from the origin because this e^3t negligible."},{"Start":"08:18.080 ","End":"08:20.120","Text":"I mentioned this before, and this one dominates,"},{"Start":"08:20.120 ","End":"08:26.774","Text":"it goes to infinity and we go along the vector minus 2,"},{"Start":"08:26.774 ","End":"08:28.730","Text":"3 which gave us this line."},{"Start":"08:28.730 ","End":"08:33.560","Text":"That\u0027s the one going diagonally this way."},{"Start":"08:33.560 ","End":"08:37.520","Text":"That\u0027s about it."},{"Start":"08:37.520 ","End":"08:41.610","Text":"That\u0027s all I want to say."}],"ID":10710},{"Watched":false,"Name":"exercise 3","Duration":"7m 37s","ChapterTopicVideoID":10346,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.315","Text":"In this exercise, we have a system of differential equations,"},{"Start":"00:06.315 ","End":"00:08.610","Text":"planar, meaning in 2D."},{"Start":"00:08.610 ","End":"00:10.590","Text":"It\u0027s from the world of physics,"},{"Start":"00:10.590 ","End":"00:15.105","Text":"where x describes the position of a particle"},{"Start":"00:15.105 ","End":"00:21.120","Text":"and the system of equations is that x\u0027 is Ax,"},{"Start":"00:21.120 ","End":"00:23.175","Text":"where a is a 2 by 2 matrix,"},{"Start":"00:23.175 ","End":"00:24.810","Text":"that\u0027s written here,"},{"Start":"00:24.810 ","End":"00:29.655","Text":"and x is a vector variable dependent on time,"},{"Start":"00:29.655 ","End":"00:35.625","Text":"it\u0027s the position of the particle in the plane at time t. 2 questions,"},{"Start":"00:35.625 ","End":"00:43.980","Text":"a, find the general solution for this system of differential equations."},{"Start":"00:45.530 ","End":"00:50.480","Text":"We\u0027re going to whiz through this one because the more important one is b,"},{"Start":"00:50.480 ","End":"00:52.925","Text":"which is to classify the critical point."},{"Start":"00:52.925 ","End":"00:56.435","Text":"There\u0027s only going to be 1 critical point, it\u0027s the origin."},{"Start":"00:56.435 ","End":"01:02.735","Text":"To sketch the phase plane portrait of the system."},{"Start":"01:02.735 ","End":"01:05.630","Text":"This is what we\u0027re concentrating on."},{"Start":"01:05.630 ","End":"01:09.020","Text":"Like I said, we\u0027re just going to whiz through part a."},{"Start":"01:09.020 ","End":"01:12.680","Text":"Take the matrix, you find the eigenvalues the usual way."},{"Start":"01:12.680 ","End":"01:16.270","Text":"They come out to be minus 2 and 8."},{"Start":"01:16.270 ","End":"01:21.850","Text":"The corresponding eigenvectors come out to be minus 1,"},{"Start":"01:21.850 ","End":"01:25.105","Text":"2 and 3, 4."},{"Start":"01:25.105 ","End":"01:29.230","Text":"This goes with this, this one goes with this one."},{"Start":"01:29.230 ","End":"01:33.580","Text":"Then we build a fundamental set of solutions,"},{"Start":"01:33.580 ","End":"01:35.200","Text":"it\u0027s going to be 2 of them."},{"Start":"01:35.200 ","End":"01:39.625","Text":"The first 1 we can get just by e^minus the Lambda,"},{"Start":"01:39.625 ","End":"01:45.250","Text":"the first eigenvalue t and then the first eigenvector and"},{"Start":"01:45.250 ","End":"01:52.220","Text":"similarly to the second eigenvalue times t and the second eigenvector."},{"Start":"01:52.560 ","End":"02:00.220","Text":"Then the general solution is just a linear combination of these 2,"},{"Start":"02:00.220 ","End":"02:01.615","Text":"c_1 times this one,"},{"Start":"02:01.615 ","End":"02:04.655","Text":"c_2 times the other one."},{"Start":"02:04.655 ","End":"02:09.460","Text":"That was part a. Now on to part b,"},{"Start":"02:09.460 ","End":"02:12.680","Text":"where we have to find the critical points."},{"Start":"02:12.680 ","End":"02:17.885","Text":"But we know with this kind of equation that it\u0027s just the origin,"},{"Start":"02:17.885 ","End":"02:21.395","Text":"at least when the determinant of A is not 0,"},{"Start":"02:21.395 ","End":"02:23.195","Text":"A is a non-singular matrix."},{"Start":"02:23.195 ","End":"02:26.090","Text":"If you go back and check the determinant,"},{"Start":"02:26.090 ","End":"02:29.315","Text":"it\u0027s not 0, actually we could go back and look at it."},{"Start":"02:29.315 ","End":"02:33.005","Text":"Determinant 4 times 2 minus 8 times 3."},{"Start":"02:33.005 ","End":"02:35.970","Text":"Well, it\u0027s not 0, of course."},{"Start":"02:36.650 ","End":"02:41.835","Text":"The only solution is 0, 0."},{"Start":"02:41.835 ","End":"02:44.444","Text":"That\u0027s the origin."},{"Start":"02:44.444 ","End":"02:46.980","Text":"Now the classification."},{"Start":"02:46.980 ","End":"02:51.110","Text":"Well, the eigenvalues tell us that when we have"},{"Start":"02:51.110 ","End":"02:55.715","Text":"2 different eigenvalues with opposite signs,"},{"Start":"02:55.715 ","End":"02:57.800","Text":"a negative and a positive,"},{"Start":"02:57.800 ","End":"03:03.020","Text":"then it\u0027s always a saddle point and a saddle point\u0027s always unstable."},{"Start":"03:03.020 ","End":"03:06.270","Text":"That was that."},{"Start":"03:06.880 ","End":"03:09.530","Text":"We still have to do a sketch."},{"Start":"03:09.530 ","End":"03:12.049","Text":"We\u0027ll do that on the next page."},{"Start":"03:12.049 ","End":"03:20.100","Text":"Here we are again and I just copied the general solution and emphasized the eigenvectors."},{"Start":"03:21.920 ","End":"03:27.870","Text":"I think I\u0027ll just show you the final sketch and then explain how we got to it."},{"Start":"03:29.260 ","End":"03:32.700","Text":"This is what it looks like."},{"Start":"03:33.530 ","End":"03:37.580","Text":"I\u0027ll explain now the steps."},{"Start":"03:37.580 ","End":"03:41.840","Text":"The first step is to get these 2 lines here."},{"Start":"03:41.840 ","End":"03:47.210","Text":"These are based on these 2 vectors which we just extend,"},{"Start":"03:47.210 ","End":"03:51.680","Text":"take the line through each of these that passes through the origin."},{"Start":"03:51.680 ","End":"03:53.870","Text":"This may not be exactly to scale,"},{"Start":"03:53.870 ","End":"04:00.350","Text":"but let\u0027s say this was minus 1, 2."},{"Start":"04:00.350 ","End":"04:04.950","Text":"Then we would take also, maybe,"},{"Start":"04:04.950 ","End":"04:07.310","Text":"I don\u0027t know where it is, but 3,"},{"Start":"04:07.310 ","End":"04:11.710","Text":"4 and then there\u0027s the origin."},{"Start":"04:12.070 ","End":"04:16.160","Text":"Now I want to explain about the arrows."},{"Start":"04:16.160 ","End":"04:20.450","Text":"This one and this one."},{"Start":"04:20.450 ","End":"04:23.015","Text":"If c_2 is 0,"},{"Start":"04:23.015 ","End":"04:27.660","Text":"meaning we just have the multiples of minus 1,"},{"Start":"04:27.660 ","End":"04:30.025","Text":"2, there\u0027s 2 cases."},{"Start":"04:30.025 ","End":"04:32.750","Text":"The e to the power of, is always positive."},{"Start":"04:32.750 ","End":"04:35.315","Text":"If c_1 is positive,"},{"Start":"04:35.315 ","End":"04:40.850","Text":"then we\u0027re in the second quadrant and as t goes to infinity,"},{"Start":"04:40.850 ","End":"04:45.400","Text":"this goes to 0, so we\u0027re going towards the origin."},{"Start":"04:45.400 ","End":"04:48.305","Text":"If c_1 is negative,"},{"Start":"04:48.305 ","End":"04:51.530","Text":"then we\u0027re in the fourth quadrant, but again,"},{"Start":"04:51.530 ","End":"04:53.675","Text":"when t goes to infinity,"},{"Start":"04:53.675 ","End":"04:55.530","Text":"we\u0027re going towards 0,"},{"Start":"04:55.530 ","End":"04:58.605","Text":"so the arrows go this way and this way."},{"Start":"04:58.605 ","End":"05:01.290","Text":"If c_1 is 0,"},{"Start":"05:01.290 ","End":"05:03.145","Text":"this 8 is positive."},{"Start":"05:03.145 ","End":"05:05.060","Text":"When t goes to infinity,"},{"Start":"05:05.060 ","End":"05:07.205","Text":"this goes to infinity,"},{"Start":"05:07.205 ","End":"05:13.470","Text":"they go away from the origin towards the infinity."},{"Start":"05:14.330 ","End":"05:20.820","Text":"That\u0027s that and that and that explains the arrows."},{"Start":"05:21.130 ","End":"05:25.490","Text":"Now we\u0027re discussing when c_1 and c_2 are not 0,"},{"Start":"05:25.490 ","End":"05:29.685","Text":"we\u0027re off these 2 diagonal lines."},{"Start":"05:29.685 ","End":"05:35.330","Text":"We look what happens when t goes to infinity and then when t goes to minus infinity,"},{"Start":"05:35.330 ","End":"05:36.875","Text":"so first the infinity."},{"Start":"05:36.875 ","End":"05:41.095","Text":"Now the trajectories move away from the origin."},{"Start":"05:41.095 ","End":"05:45.185","Text":"They get farther and farther away when t goes to infinity."},{"Start":"05:45.185 ","End":"05:48.035","Text":"Now, remember it was made up of 2 bits."},{"Start":"05:48.035 ","End":"05:53.750","Text":"The e^minus 2t negligible goes to 0,"},{"Start":"05:53.750 ","End":"05:56.980","Text":"so really the e^8t counts."},{"Start":"05:56.980 ","End":"06:00.830","Text":"When t goes to infinity, it\u0027s like we can just cross this one off and it"},{"Start":"06:00.830 ","End":"06:04.670","Text":"behaves like this so it goes to infinity."},{"Start":"06:04.670 ","End":"06:06.965","Text":"Since this becomes negligible,"},{"Start":"06:06.965 ","End":"06:11.127","Text":"we\u0027re approaching the line that goes through 3,"},{"Start":"06:11.127 ","End":"06:14.910","Text":"4, which is the line x_2."},{"Start":"06:14.910 ","End":"06:21.390","Text":"For example, this arrow and let\u0027s see."},{"Start":"06:21.390 ","End":"06:25.870","Text":"This arrow here, everything that goes to, as we get to,"},{"Start":"06:25.870 ","End":"06:27.560","Text":"as t goes to infinity,"},{"Start":"06:27.560 ","End":"06:29.720","Text":"we\u0027re going asymptotically towards this line,"},{"Start":"06:29.720 ","End":"06:31.565","Text":"either on this side or on this side."},{"Start":"06:31.565 ","End":"06:36.290","Text":"Similarly this arrow and what"},{"Start":"06:36.290 ","End":"06:41.510","Text":"else on this arrow would be another example of going towards this line?"},{"Start":"06:41.510 ","End":"06:43.805","Text":"On the other hand,"},{"Start":"06:43.805 ","End":"06:46.459","Text":"when t goes to minus infinity,"},{"Start":"06:46.459 ","End":"06:50.570","Text":"it\u0027s just the opposite that this thing b goes to infinity."},{"Start":"06:50.570 ","End":"06:56.240","Text":"This becomes negligible and we are asymptotic to the other line,"},{"Start":"06:56.240 ","End":"07:02.975","Text":"which was the line x_2 equals minus 2x_1 and that would explain."},{"Start":"07:02.975 ","End":"07:05.185","Text":"Let me use a different color."},{"Start":"07:05.185 ","End":"07:08.960","Text":"Let\u0027s say this, going further back."},{"Start":"07:08.960 ","End":"07:10.430","Text":"If we go to minus infinity,"},{"Start":"07:10.430 ","End":"07:13.850","Text":"we\u0027re coming from this line,"},{"Start":"07:13.850 ","End":"07:17.529","Text":"coming from this line."},{"Start":"07:17.529 ","End":"07:18.950","Text":"In the other direction also,"},{"Start":"07:18.950 ","End":"07:27.270","Text":"we\u0027re coming from the line x_1,"},{"Start":"07:27.270 ","End":"07:30.280","Text":"which is x_2 equals minus 2x_1."},{"Start":"07:32.450 ","End":"07:37.810","Text":"That\u0027s basically it and I guess we\u0027re done."}],"ID":10711},{"Watched":false,"Name":"exercise 4","Duration":"7m 16s","ChapterTopicVideoID":10347,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:02.819","Text":"Here, we have another one of those exercises,"},{"Start":"00:02.819 ","End":"00:05.310","Text":"where we have a planar system."},{"Start":"00:05.310 ","End":"00:12.405","Text":"By planar, I mean 2D of the linear differential equations,"},{"Start":"00:12.405 ","End":"00:15.760","Text":"constant coefficients, homogeneous."},{"Start":"00:16.220 ","End":"00:22.140","Text":"It\u0027s given in the matrix form where x\u0027 is a times x and x is,"},{"Start":"00:22.140 ","End":"00:25.185","Text":"we take it as a column matrix 2-dimensional, x_1,"},{"Start":"00:25.185 ","End":"00:28.590","Text":"x_2 as a function of t, is some parameter."},{"Start":"00:28.590 ","End":"00:32.760","Text":"In physics, it\u0027s usually time. We have 2 things."},{"Start":"00:32.760 ","End":"00:37.940","Text":"Find the general solution and to classify the critical point."},{"Start":"00:37.940 ","End":"00:39.050","Text":"I say, the critical point,"},{"Start":"00:39.050 ","End":"00:44.510","Text":"because this system always has a critical point at the origin, and that\u0027s the only one."},{"Start":"00:44.510 ","End":"00:52.130","Text":"The second part of b, is to give a rough sketch of the phase portrait it."},{"Start":"00:52.130 ","End":"00:54.420","Text":"Let\u0027s start with part a."},{"Start":"00:54.420 ","End":"00:56.870","Text":"We\u0027re going to whiz through this part because really,"},{"Start":"00:56.870 ","End":"00:59.089","Text":"b is the one I\u0027m emphasizing."},{"Start":"00:59.089 ","End":"01:01.670","Text":"It\u0027s the sketching the portrait."},{"Start":"01:01.670 ","End":"01:04.610","Text":"First step is to find the eigenvalues."},{"Start":"01:04.610 ","End":"01:09.030","Text":"This time, they came out not to be real numbers."},{"Start":"01:09.030 ","End":"01:11.495","Text":"It\u0027s always comes out in conjugate pairs,"},{"Start":"01:11.495 ","End":"01:13.145","Text":"if this is a real matrix,"},{"Start":"01:13.145 ","End":"01:16.620","Text":"1 plus 2i and 1 minus 2i."},{"Start":"01:16.720 ","End":"01:23.555","Text":"The next step is to find the eigenvectors corresponding to each eigenvalue."},{"Start":"01:23.555 ","End":"01:29.720","Text":"It always turns out that one is the complex conjugate of the other."},{"Start":"01:29.720 ","End":"01:32.015","Text":"You see, i here and minus i."},{"Start":"01:32.015 ","End":"01:37.700","Text":"Now, we can easily get a fundamental set of solutions with complex numbers,"},{"Start":"01:37.700 ","End":"01:39.530","Text":"just like we usually do."},{"Start":"01:39.530 ","End":"01:46.205","Text":"We take e to the power of 1 eigenvalue t times its eigenvector, and then the other."},{"Start":"01:46.205 ","End":"01:49.550","Text":"But we don\u0027t want complex solutions."},{"Start":"01:49.550 ","End":"01:51.905","Text":"We want real solutions."},{"Start":"01:51.905 ","End":"01:57.050","Text":"The way we get to the real solutions is to just pick one of these 2."},{"Start":"01:57.050 ","End":"01:59.180","Text":"It doesn\u0027t matter, say x^1,"},{"Start":"01:59.180 ","End":"02:03.755","Text":"and take the real and imaginary parts of it."},{"Start":"02:03.755 ","End":"02:10.265","Text":"We use Euler\u0027s formula to expand e to the power of a complex number."},{"Start":"02:10.265 ","End":"02:13.824","Text":"It\u0027s e to the power of the real part."},{"Start":"02:13.824 ","End":"02:20.265","Text":"Then cosine plus i sine of the imaginary part."},{"Start":"02:20.265 ","End":"02:22.215","Text":"This part here is this,"},{"Start":"02:22.215 ","End":"02:24.345","Text":"and there\u0027s the vector."},{"Start":"02:24.345 ","End":"02:28.580","Text":"Let\u0027s give names to the real and the imaginary parts,"},{"Start":"02:28.580 ","End":"02:33.860","Text":"called them u and v. What we do is we pick out the real part."},{"Start":"02:33.860 ","End":"02:38.900","Text":"Now, for the top component,"},{"Start":"02:38.900 ","End":"02:41.335","Text":"for the x_1 component,"},{"Start":"02:41.335 ","End":"02:44.090","Text":"we\u0027re going to take the real part here,"},{"Start":"02:44.090 ","End":"02:47.460","Text":"which is e^t cosine 2t."},{"Start":"02:47.500 ","End":"02:51.495","Text":"The real part for the x_2 component,"},{"Start":"02:51.495 ","End":"02:52.965","Text":"we need i,"},{"Start":"02:52.965 ","End":"02:55.145","Text":"because i with i will give a real number."},{"Start":"02:55.145 ","End":"02:57.170","Text":"We take this with this,"},{"Start":"02:57.170 ","End":"03:03.390","Text":"and we get I^2 sine 2t with e^t and i^2 is minus 1."},{"Start":"03:03.390 ","End":"03:06.120","Text":"Then similarly, for v,"},{"Start":"03:06.120 ","End":"03:08.025","Text":"one of the imaginary part."},{"Start":"03:08.025 ","End":"03:10.430","Text":"At the top, we want from here."},{"Start":"03:10.430 ","End":"03:13.315","Text":"At the bottom, we want this bit."},{"Start":"03:13.315 ","End":"03:19.399","Text":"These are our u and v. Then we get the general solution"},{"Start":"03:19.399 ","End":"03:25.220","Text":"by taking some linear combination of these 2,"},{"Start":"03:25.220 ","End":"03:29.980","Text":"so c_1 times this plus c_2 times this."},{"Start":"03:29.980 ","End":"03:33.150","Text":"That answers part A."},{"Start":"03:33.150 ","End":"03:38.270","Text":"Part b, we already mentioned that there\u0027s only one critical point,"},{"Start":"03:38.270 ","End":"03:41.810","Text":"which is at the origin because the matrix is non-singular,"},{"Start":"03:41.810 ","End":"03:45.080","Text":"and we have to classify it."},{"Start":"03:45.080 ","End":"03:48.655","Text":"The eigenvalues are complex."},{"Start":"03:48.655 ","End":"03:52.215","Text":"Remember, it was 1 plus or minus 2i."},{"Start":"03:52.215 ","End":"04:01.040","Text":"Now, when it\u0027s a complex conjugate pair like this with a non-zero real part,"},{"Start":"04:01.040 ","End":"04:04.200","Text":"then it\u0027s a spiral point."},{"Start":"04:04.250 ","End":"04:08.935","Text":"Depending on whether this real part is positive or negative,"},{"Start":"04:08.935 ","End":"04:11.095","Text":"it will be unstable,"},{"Start":"04:11.095 ","End":"04:13.120","Text":"or if it was negative,"},{"Start":"04:13.120 ","End":"04:15.855","Text":"it would be asymptotically stable."},{"Start":"04:15.855 ","End":"04:21.685","Text":"Based on this, we can say that it\u0027s spiral and unstable."},{"Start":"04:21.685 ","End":"04:25.585","Text":"The next thing we want to do is the sketch."},{"Start":"04:25.585 ","End":"04:29.290","Text":"I just brought this in as a reminder."},{"Start":"04:29.290 ","End":"04:33.710","Text":"I think, it\u0027s best, I\u0027ll just present it and then explain it."},{"Start":"04:33.710 ","End":"04:38.020","Text":"Here\u0027s what a spiral generally looks like."},{"Start":"04:38.020 ","End":"04:40.930","Text":"There\u0027s 2 main things to discover,"},{"Start":"04:40.930 ","End":"04:46.865","Text":"whether it goes outwards or inwards."},{"Start":"04:46.865 ","End":"04:52.660","Text":"The second thing is whether it goes clockwise or counterclockwise."},{"Start":"04:52.660 ","End":"04:57.670","Text":"We already mentioned that it\u0027s unstable because of the positive real part,"},{"Start":"04:57.670 ","End":"05:02.410","Text":"which makes this e^t go to infinity."},{"Start":"05:02.410 ","End":"05:06.060","Text":"We know that the arrows are going outward."},{"Start":"05:06.060 ","End":"05:11.370","Text":"Question now, is clockwise or counterclockwise?"},{"Start":"05:12.100 ","End":"05:16.135","Text":"I just wrote down some of what I just said."},{"Start":"05:16.135 ","End":"05:20.135","Text":"How do we discover which direction it\u0027s going in?"},{"Start":"05:20.135 ","End":"05:23.940","Text":"Now, this diagram is just one of the spirals."},{"Start":"05:23.940 ","End":"05:25.230","Text":"There\u0027s an infinite number."},{"Start":"05:25.230 ","End":"05:27.570","Text":"You vary c_1 and c_2."},{"Start":"05:27.570 ","End":"05:31.985","Text":"But it turns out that all the spirals are of the same nature."},{"Start":"05:31.985 ","End":"05:34.175","Text":"They don\u0027t cut each other."},{"Start":"05:34.175 ","End":"05:38.685","Text":"The orbits can\u0027t cut each other at trajectories."},{"Start":"05:38.685 ","End":"05:43.845","Text":"They\u0027re all going to be clockwise or all counterclockwise."},{"Start":"05:43.845 ","End":"05:45.800","Text":"We just pick a convenient one."},{"Start":"05:45.800 ","End":"05:47.585","Text":"Let\u0027s say u of t,"},{"Start":"05:47.585 ","End":"05:53.945","Text":"which is when we set c_1 is 1 and c_2 is 0, for example."},{"Start":"05:53.945 ","End":"05:59.165","Text":"Let\u0027s see if I can go back and see where u of t is. Here it is."},{"Start":"05:59.165 ","End":"06:04.190","Text":"Let\u0027s plug in a value easiest to let t=0,"},{"Start":"06:04.190 ","End":"06:08.715","Text":"and then we\u0027ll get cosine of 0 minus sine 0."},{"Start":"06:08.715 ","End":"06:14.540","Text":"We\u0027ll get the point 1, 0."},{"Start":"06:14.540 ","End":"06:19.320","Text":"Let me highlight it."},{"Start":"06:19.320 ","End":"06:22.810","Text":"Let\u0027s say, this is the point 1, 0."},{"Start":"06:23.560 ","End":"06:26.555","Text":"Then what we want to do,"},{"Start":"06:26.555 ","End":"06:28.400","Text":"is take the direction,"},{"Start":"06:28.400 ","End":"06:31.385","Text":"the tangent, the x\u0027 at that point."},{"Start":"06:31.385 ","End":"06:36.810","Text":"We can do that using the formula that x\u0027 is A times x."},{"Start":"06:36.810 ","End":"06:40.650","Text":"When you let t=0,"},{"Start":"06:40.650 ","End":"06:44.730","Text":"we get that x is 1,"},{"Start":"06:44.730 ","End":"06:47.610","Text":"0, multiply that by the matrix A,"},{"Start":"06:47.610 ","End":"06:50.145","Text":"and it comes out to be 1, 2."},{"Start":"06:50.145 ","End":"06:51.765","Text":"Let me just go back a bit."},{"Start":"06:51.765 ","End":"06:53.550","Text":"The vector 1,"},{"Start":"06:53.550 ","End":"06:58.475","Text":"2 would be something in this direction."},{"Start":"06:58.475 ","End":"07:02.060","Text":"Anyway, it has an upward component, so obviously,"},{"Start":"07:02.060 ","End":"07:07.110","Text":"we are going to be going counterclockwise."},{"Start":"07:09.400 ","End":"07:13.910","Text":"Really, that\u0027s all there is to it."},{"Start":"07:13.910 ","End":"07:16.290","Text":"I think, we\u0027ll stop here."}],"ID":10712},{"Watched":false,"Name":"exercise 5","Duration":"8m 56s","ChapterTopicVideoID":10348,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.465","Text":"In this exercise, we have a non-linear planar system of differential equations,"},{"Start":"00:06.465 ","End":"00:09.120","Text":"dx over dt and dy over dt."},{"Start":"00:09.120 ","End":"00:13.455","Text":"Note that both right-hand sides consists of just x and y,"},{"Start":"00:13.455 ","End":"00:16.095","Text":"t doesn\u0027t appear in them."},{"Start":"00:16.095 ","End":"00:21.435","Text":"These are sometimes called autonomous systems when this is so."},{"Start":"00:21.435 ","End":"00:26.535","Text":"We have several tasks to find the critical points."},{"Start":"00:26.535 ","End":"00:32.115","Text":"Near each critical point to find the corresponding linear system."},{"Start":"00:32.115 ","End":"00:36.030","Text":"Then we have to determine its type."},{"Start":"00:36.030 ","End":"00:44.000","Text":"Then the stability of the original system,"},{"Start":"00:44.000 ","End":"00:45.950","Text":"not the corresponding linear system,"},{"Start":"00:45.950 ","End":"00:48.590","Text":"but it turns out to be the same."},{"Start":"00:48.590 ","End":"00:56.599","Text":"Then we\u0027re going to try and give a rough sketch of the phase portrait for the system."},{"Start":"00:56.599 ","End":"01:00.410","Text":"The last part is that we assume that x and y,"},{"Start":"01:00.410 ","End":"01:06.710","Text":"which are non-negative, represent populations of certain species."},{"Start":"01:06.710 ","End":"01:11.280","Text":"This is a differential equation that governs them."},{"Start":"01:11.390 ","End":"01:17.000","Text":"We\u0027re asking, is there any way that these species could co-exist in"},{"Start":"01:17.000 ","End":"01:24.360","Text":"a stable way if they satisfy the governed by these equations."},{"Start":"01:24.850 ","End":"01:29.330","Text":"We\u0027ll start with Part A."},{"Start":"01:29.330 ","End":"01:36.395","Text":"Remember that a critical point is where dx by dt and dy by dt are both 0,"},{"Start":"01:36.395 ","End":"01:39.875","Text":"and dx by dt is this and dy by dt is this."},{"Start":"01:39.875 ","End":"01:47.010","Text":"We have an algebraic system of 2 equations in 2 unknowns."},{"Start":"01:47.010 ","End":"01:51.290","Text":"I won\u0027t waste time going through all the details of the algebra."},{"Start":"01:51.290 ","End":"01:53.090","Text":"This is presented here."},{"Start":"01:53.090 ","End":"01:55.970","Text":"We basically get 4 solutions for x,"},{"Start":"01:55.970 ","End":"01:57.665","Text":"y for this system,"},{"Start":"01:57.665 ","End":"02:04.960","Text":"and these that I\u0027ve colored the 4 solutions to this algebraic system."},{"Start":"02:04.960 ","End":"02:07.470","Text":"Now that we\u0027ve found the 4 critical points,"},{"Start":"02:07.470 ","End":"02:09.825","Text":"let\u0027s move on to Part B."},{"Start":"02:09.825 ","End":"02:14.225","Text":"Let\u0027s give these 2 functions names."},{"Start":"02:14.225 ","End":"02:19.020","Text":"Let this one be big F(x,y),"},{"Start":"02:19.020 ","End":"02:22.455","Text":"and this one we\u0027ll call big G(x,y),"},{"Start":"02:22.455 ","End":"02:27.275","Text":"and I will also expand the brackets multiplied x here and multiplied y here."},{"Start":"02:27.275 ","End":"02:30.035","Text":"This is the system we have."},{"Start":"02:30.035 ","End":"02:34.630","Text":"From this we get the Jacobian matrix,"},{"Start":"02:34.630 ","End":"02:38.560","Text":"which in general is the matrix of"},{"Start":"02:38.560 ","End":"02:45.155","Text":"partial derivatives of f and g with respect to x and y, placed like this."},{"Start":"02:45.155 ","End":"02:48.640","Text":"If you just do the computations of the partial derivatives,"},{"Start":"02:48.640 ","End":"02:51.324","Text":"for example, df by dx,"},{"Start":"02:51.324 ","End":"02:59.625","Text":"then y is a constant and we just get 4 minus 2x and here minus 3y."},{"Start":"02:59.625 ","End":"03:01.450","Text":"Then similarly for the rest of them,"},{"Start":"03:01.450 ","End":"03:03.670","Text":"that\u0027s the Jacobian matrix,"},{"Start":"03:03.670 ","End":"03:06.490","Text":"but it depends on x, y."},{"Start":"03:06.490 ","End":"03:09.655","Text":"We plug in each of the critical points,"},{"Start":"03:09.655 ","End":"03:13.240","Text":"and then that gives us a matrix of constants."},{"Start":"03:13.240 ","End":"03:20.300","Text":"The system behaves like a linear system at each of these critical points."},{"Start":"03:20.300 ","End":"03:30.260","Text":"Then we get the eigenvalues of these matrices and according to the signs and so on,"},{"Start":"03:30.260 ","End":"03:33.260","Text":"we can determine the type and stability."},{"Start":"03:33.260 ","End":"03:38.030","Text":"Here we are on the new page and this is the Jacobian and we\u0027re"},{"Start":"03:38.030 ","End":"03:42.680","Text":"about to plug-in the 4 critical points, one at a time."},{"Start":"03:42.680 ","End":"03:43.910","Text":"The first one was 0,"},{"Start":"03:43.910 ","End":"03:45.680","Text":"0, plugging it in here."},{"Start":"03:45.680 ","End":"03:48.920","Text":"This is the matrix we get its diagonal,"},{"Start":"03:48.920 ","End":"03:52.175","Text":"so clearly the eigenvalues are 4 and 3."},{"Start":"03:52.175 ","End":"03:58.415","Text":"If you look it up, this one real positive,"},{"Start":"03:58.415 ","End":"04:03.090","Text":"different eigenvalues is an unstable node."},{"Start":"04:03.400 ","End":"04:06.605","Text":"Next we plug in 0,"},{"Start":"04:06.605 ","End":"04:08.660","Text":"1 1/2 into here."},{"Start":"04:08.660 ","End":"04:12.385","Text":"We get this, the eigenvalues for this one,"},{"Start":"04:12.385 ","End":"04:14.210","Text":"well, when there\u0027s a 0 on the diagonal,"},{"Start":"04:14.210 ","End":"04:20.165","Text":"it\u0027s just these 2 are the eigenvalues and they\u0027re both real negative different,"},{"Start":"04:20.165 ","End":"04:26.850","Text":"so that is asymptotically stable node."},{"Start":"04:26.920 ","End":"04:31.160","Text":"When they both have the same sign, it\u0027s a node."},{"Start":"04:31.160 ","End":"04:32.690","Text":"If they\u0027re both positive,"},{"Start":"04:32.690 ","End":"04:37.660","Text":"it\u0027s unstable, both negative, stable asymptotically."},{"Start":"04:37.660 ","End":"04:40.985","Text":"The next one we plug-in 4, 0."},{"Start":"04:40.985 ","End":"04:47.870","Text":"Again, we get 2 negative different real eigenvalues."},{"Start":"04:47.870 ","End":"04:51.920","Text":"Again, an asymptotically stable node."},{"Start":"04:51.920 ","End":"04:56.360","Text":"The last one, it was 1,"},{"Start":"04:56.360 ","End":"05:00.490","Text":"1, plug it in,"},{"Start":"05:00.490 ","End":"05:04.980","Text":"and we get this matrix."},{"Start":"05:04.980 ","End":"05:09.110","Text":"If you do the computation, I\u0027m not going to do it."},{"Start":"05:09.110 ","End":"05:14.824","Text":"The eigenvalues come out to be real with opposite signs."},{"Start":"05:14.824 ","End":"05:22.400","Text":"You can see they are opposite signs because the square root of 13 is bigger than 3,"},{"Start":"05:22.400 ","End":"05:25.790","Text":"so when I subtract 3 minus root 13,"},{"Start":"05:25.790 ","End":"05:30.260","Text":"I\u0027ll get a negative eigenvalue and with the plus get a positive eigenvalue."},{"Start":"05:30.260 ","End":"05:36.295","Text":"Opposite signs gives us a saddle and a saddle is always unstable."},{"Start":"05:36.295 ","End":"05:41.880","Text":"Now, near the critical points of the original system."},{"Start":"05:43.070 ","End":"05:46.430","Text":"This linear system approximates"},{"Start":"05:46.430 ","End":"05:53.100","Text":"the original system in the sense that they\u0027re going to be the same kinds of nodes."},{"Start":"05:55.000 ","End":"05:59.810","Text":"The ones that are unstable, this one,"},{"Start":"05:59.810 ","End":"06:02.540","Text":"and where else does it say unstable,"},{"Start":"06:02.540 ","End":"06:05.549","Text":"there, unstable, unstable."},{"Start":"06:05.549 ","End":"06:07.360","Text":"That\u0027s the 0,"},{"Start":"06:07.360 ","End":"06:08.790","Text":"0 and 1, 1."},{"Start":"06:08.790 ","End":"06:12.465","Text":"Stable asymptotically, that\u0027s this one on this one,"},{"Start":"06:12.465 ","End":"06:14.490","Text":"so that\u0027s these 2 points 0,"},{"Start":"06:14.490 ","End":"06:18.190","Text":"3 over 2 and 4, 0."},{"Start":"06:18.500 ","End":"06:22.940","Text":"Now a sketch for this thing is pretty hard to do"},{"Start":"06:22.940 ","End":"06:26.810","Text":"unless you have, through computer aided."},{"Start":"06:26.810 ","End":"06:36.350","Text":"But I found a sketch on the Internet which is similar with the same kinds of nodes,"},{"Start":"06:36.350 ","End":"06:41.010","Text":"the saddles and stable, unstable."},{"Start":"06:41.170 ","End":"06:45.349","Text":"Now this is not a sketch of our system."},{"Start":"06:45.349 ","End":"06:48.995","Text":"It\u0027s it\u0027s a similar one."},{"Start":"06:48.995 ","End":"06:55.890","Text":"In our case like on the axis we\u0027d have the 0, 0 is fine,"},{"Start":"06:55.890 ","End":"07:00.630","Text":"but 4, 0 would be this one,"},{"Start":"07:00.630 ","End":"07:03.210","Text":"and here is 1, 0 and the 0,"},{"Start":"07:03.210 ","End":"07:05.310","Text":"3 over 2 would be this one."},{"Start":"07:05.310 ","End":"07:07.570","Text":"The numbers are different."},{"Start":"07:07.820 ","End":"07:10.995","Text":"The 1, 1"},{"Start":"07:10.995 ","End":"07:20.150","Text":"is this one here."},{"Start":"07:20.150 ","End":"07:25.147","Text":"But it\u0027s qualitatively the same idea enough to answer Part"},{"Start":"07:25.147 ","End":"07:32.750","Text":"E. We\u0027re interested in what happens to the number of each species."},{"Start":"07:32.750 ","End":"07:36.380","Text":"One of them is x and the other one is y."},{"Start":"07:36.380 ","End":"07:39.470","Text":"As time goes by and in many,"},{"Start":"07:39.470 ","End":"07:40.550","Text":"many years, in other words,"},{"Start":"07:40.550 ","End":"07:42.440","Text":"t goes to infinity."},{"Start":"07:42.440 ","End":"07:46.460","Text":"We\u0027re only interested in the stable ones."},{"Start":"07:46.460 ","End":"07:51.480","Text":"Let\u0027s see if we can go back and see the stable ones."},{"Start":"07:51.610 ","End":"07:55.910","Text":"Asymptotically stable the ones will eventually stabilize."},{"Start":"07:55.910 ","End":"08:00.860","Text":"The ones on the axis not at the origin 0,"},{"Start":"08:00.860 ","End":"08:02.600","Text":"3 over 2 and 4, 0."},{"Start":"08:02.600 ","End":"08:03.904","Text":"But in this picture,"},{"Start":"08:03.904 ","End":"08:11.330","Text":"that would be this one here and this one here."},{"Start":"08:11.330 ","End":"08:13.430","Text":"The numbers are different."},{"Start":"08:13.430 ","End":"08:18.395","Text":"Eventually these are the numbers,"},{"Start":"08:18.395 ","End":"08:21.470","Text":"of course these are not numbers of species."},{"Start":"08:21.470 ","End":"08:25.130","Text":"Maybe it\u0027s in millions of species or something."},{"Start":"08:25.130 ","End":"08:28.940","Text":"It\u0027s just the idea that matters and stuff here."},{"Start":"08:28.940 ","End":"08:34.655","Text":"Eventually they all go to this here."},{"Start":"08:34.655 ","End":"08:36.319","Text":"Now in each of these cases,"},{"Start":"08:36.319 ","End":"08:39.560","Text":"eventually we have just y\u0027s and no,"},{"Start":"08:39.560 ","End":"08:43.310","Text":"x\u0027s are just x\u0027s and no y\u0027s."},{"Start":"08:43.310 ","End":"08:47.299","Text":"Because on the axis one of the variables is 0,"},{"Start":"08:47.299 ","End":"08:52.820","Text":"which means that only one of the species survive and the other tends to 0."},{"Start":"08:52.820 ","End":"08:57.930","Text":"That answers all the parts of the question and we are done."}],"ID":10713},{"Watched":false,"Name":"exercise 6","Duration":"6m 32s","ChapterTopicVideoID":10349,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.050 ","End":"00:06.300","Text":"In this exercise, we\u0027re told that A is a 2 by 2 real matrix,"},{"Start":"00:06.300 ","End":"00:08.745","Text":"but we\u0027re not given the matrix A."},{"Start":"00:08.745 ","End":"00:12.240","Text":"We\u0027re told that it has these 2 eigenvalues,"},{"Start":"00:12.240 ","End":"00:13.620","Text":"minus 1 and minus 5,"},{"Start":"00:13.620 ","End":"00:18.075","Text":"and that the corresponding eigenvectors are this and this."},{"Start":"00:18.075 ","End":"00:23.360","Text":"We have to give a general solution to the linear planar system,"},{"Start":"00:23.360 ","End":"00:27.230","Text":"x\u0027=Ax, even though we don\u0027t know A,"},{"Start":"00:27.230 ","End":"00:30.655","Text":"we just have these hints about it."},{"Start":"00:30.655 ","End":"00:35.440","Text":"The vector x is a 2D vector, x_1 and x_2."},{"Start":"00:35.830 ","End":"00:38.015","Text":"Then in b,"},{"Start":"00:38.015 ","End":"00:43.670","Text":"we\u0027re asked to find an invertible matrix T and a diagonal matrix D such"},{"Start":"00:43.670 ","End":"00:51.410","Text":"that A is similar to D in the sense that there is an invertible matrix T,"},{"Start":"00:51.410 ","End":"00:57.150","Text":"so T minus 1AT is D. Then we have to"},{"Start":"00:57.150 ","End":"01:03.590","Text":"compute the exponent of t times the matrix A."},{"Start":"01:03.590 ","End":"01:05.630","Text":"Finally, to sketch"},{"Start":"01:05.630 ","End":"01:15.215","Text":"a phase portrait and to classify the type and stability of the critical point,"},{"Start":"01:15.215 ","End":"01:18.660","Text":"which is the origin where A is non-singular, of course,"},{"Start":"01:18.660 ","End":"01:22.010","Text":"it\u0027s non-singular because it has 2 different eigenvalues."},{"Start":"01:22.010 ","End":"01:23.720","Text":"Let\u0027s start with a then."},{"Start":"01:23.720 ","End":"01:30.656","Text":"We\u0027ve got e^-1t times this eigenvector,"},{"Start":"01:30.656 ","End":"01:35.900","Text":"that\u0027s here, then e^-5t times this, that\u0027s here."},{"Start":"01:35.900 ","End":"01:38.045","Text":"Once we have a fundamental set,"},{"Start":"01:38.045 ","End":"01:44.595","Text":"linear combination gives us the general solution."},{"Start":"01:44.595 ","End":"01:46.980","Text":"The fundamental matrix,"},{"Start":"01:46.980 ","End":"01:48.775","Text":"and that\u0027s going to be our T,"},{"Start":"01:48.775 ","End":"01:56.880","Text":"is simply taken by taking the columns of this matrix from the eigenvector,"},{"Start":"01:56.880 ","End":"02:00.770","Text":"so its 1, 3 is the first column and then minus 1, 1."},{"Start":"02:00.770 ","End":"02:04.990","Text":"Then we just do a computation of T inverse."},{"Start":"02:04.990 ","End":"02:07.685","Text":"For 2 by 2 matrix it\u0027s easy to do."},{"Start":"02:07.685 ","End":"02:13.160","Text":"You exchange the position of these 2,"},{"Start":"02:13.160 ","End":"02:16.400","Text":"this diagonal, make these 2 negative,"},{"Start":"02:16.400 ","End":"02:17.690","Text":"what they were,"},{"Start":"02:17.690 ","End":"02:20.390","Text":"and then divide by the determinant,"},{"Start":"02:20.390 ","End":"02:23.865","Text":"which is 1 times 1 minus minus 3,"},{"Start":"02:23.865 ","End":"02:27.250","Text":"which is 4/3, 1/4 of this."},{"Start":"02:27.860 ","End":"02:37.387","Text":"The diagonal matrix is simply the matrix containing the eigenvalues, where were they?"},{"Start":"02:37.387 ","End":"02:40.260","Text":"We can see them here, minus 1 and minus 5."},{"Start":"02:40.260 ","End":"02:42.109","Text":"These are the 2 eigenvalues,"},{"Start":"02:42.109 ","End":"02:43.610","Text":"I just put them on a diagonal,"},{"Start":"02:43.610 ","End":"02:45.530","Text":"and that\u0027s the diagonal matrix."},{"Start":"02:45.530 ","End":"02:47.330","Text":"Now we\u0027re going to make the computation."},{"Start":"02:47.330 ","End":"02:53.335","Text":"We\u0027re going to compute A as T times D times T minus 1."},{"Start":"02:53.335 ","End":"02:54.905","Text":"Where did I get this?"},{"Start":"02:54.905 ","End":"02:57.310","Text":"Well, if we go back,"},{"Start":"02:57.310 ","End":"03:01.770","Text":"it says that here T minus 1 AT is"},{"Start":"03:01.770 ","End":"03:06.995","Text":"D. If I just throw T on the left and T inverse on the right,"},{"Start":"03:06.995 ","End":"03:14.190","Text":"I can invert it and get A in terms of D and that would give us this equation."},{"Start":"03:14.600 ","End":"03:16.680","Text":"Just fit it all in."},{"Start":"03:16.680 ","End":"03:20.254","Text":"This was T, this was a diagonal D,"},{"Start":"03:20.254 ","End":"03:24.020","Text":"and this is T inverse, 1/4 times this."},{"Start":"03:24.020 ","End":"03:27.565","Text":"Then we need to do a bit of matrix multiplication."},{"Start":"03:27.565 ","End":"03:30.875","Text":"I\u0027ll just give you the calculation."},{"Start":"03:30.875 ","End":"03:33.955","Text":"The 1/4, you can bring out in front."},{"Start":"03:33.955 ","End":"03:40.310","Text":"I multiplied this matrix with this matrix and got this matrix."},{"Start":"03:40.310 ","End":"03:43.470","Text":"Then multiply these 2 together and you"},{"Start":"03:43.470 ","End":"03:47.270","Text":"get this as a 1/4 here and they\u0027re all divisible by 4."},{"Start":"03:47.270 ","End":"03:48.935","Text":"Divide everything by 4."},{"Start":"03:48.935 ","End":"03:52.160","Text":"Our matrix A is this,"},{"Start":"03:52.160 ","End":"03:55.830","Text":"just write it again, this is equal to our A."},{"Start":"03:55.990 ","End":"03:59.960","Text":"The question didn\u0027t ask for A, but we\u0027re going to need A for part c,"},{"Start":"03:59.960 ","End":"04:04.260","Text":"so I did it already in part b. Let\u0027s move on."},{"Start":"04:04.600 ","End":"04:09.090","Text":"In part c, we have to find e^tA."},{"Start":"04:09.920 ","End":"04:15.770","Text":"Now, this doesn\u0027t go very well directly because A is not a diagonal matrix,"},{"Start":"04:15.770 ","End":"04:17.779","Text":"we\u0027d rather have a diagonal matrix."},{"Start":"04:17.779 ","End":"04:22.235","Text":"We know that A is TDT inverse."},{"Start":"04:22.235 ","End":"04:27.590","Text":"The property of this T inverse is that you"},{"Start":"04:27.590 ","End":"04:34.295","Text":"can take it outside the exponent and put T here and T minus 1 here."},{"Start":"04:34.295 ","End":"04:42.750","Text":"Then it\u0027s easier to compute e to the power of a diagonal matrix. D was minus 1,"},{"Start":"04:42.750 ","End":"04:45.600","Text":"minus 5 on the diagonal, so multiply it by t,"},{"Start":"04:45.600 ","End":"04:48.450","Text":"so it\u0027s minus t, minus 5t on the diagonal."},{"Start":"04:48.450 ","End":"04:50.480","Text":"It\u0027s still a diagonal matrix."},{"Start":"04:50.480 ","End":"04:54.320","Text":"When you take e to the power of a diagonal matrix,"},{"Start":"04:54.320 ","End":"04:58.955","Text":"you just take e to the power of each spot here,"},{"Start":"04:58.955 ","End":"05:02.660","Text":"so it\u0027s e^-t, e^-5t."},{"Start":"05:03.550 ","End":"05:06.230","Text":"That\u0027s just a straightforward computation."},{"Start":"05:06.230 ","End":"05:08.210","Text":"e^tA, like we said here,"},{"Start":"05:08.210 ","End":"05:13.115","Text":"we just have to multiply this by t in front and T inverse in back."},{"Start":"05:13.115 ","End":"05:18.215","Text":"This one\u0027s t, this 1/4 of this thing is T inverse,"},{"Start":"05:18.215 ","End":"05:23.310","Text":"and this one\u0027s our D. The 1/4 goes out in front."},{"Start":"05:23.350 ","End":"05:29.645","Text":"This matrix times this matrix gives us this."},{"Start":"05:29.645 ","End":"05:35.020","Text":"Then this times this gives us this and I\u0027ll leave the 1/4 outside."},{"Start":"05:35.020 ","End":"05:37.095","Text":"No point in making it messier,"},{"Start":"05:37.095 ","End":"05:41.720","Text":"and that\u0027s e^tA that was requested."},{"Start":"05:42.450 ","End":"05:45.730","Text":"Now, for part d,"},{"Start":"05:45.730 ","End":"05:49.390","Text":"I want to remind you that we had this in the previous exercise."},{"Start":"05:49.390 ","End":"05:51.055","Text":"In fact, that\u0027s how I cooked it up."},{"Start":"05:51.055 ","End":"05:52.420","Text":"I took this matrix,"},{"Start":"05:52.420 ","End":"05:54.490","Text":"minus 4, 1, 3,"},{"Start":"05:54.490 ","End":"06:00.430","Text":"minus 2, from a previous exercise which we already sketched."},{"Start":"06:00.430 ","End":"06:08.440","Text":"We know how to classify the origin because it has 2 negative real eigenvalues,"},{"Start":"06:08.440 ","End":"06:10.735","Text":"minus 1 and minus 5."},{"Start":"06:10.735 ","End":"06:17.165","Text":"It\u0027s a node point and it\u0027s asymptotically stable."},{"Start":"06:17.165 ","End":"06:18.560","Text":"If they were both positive,"},{"Start":"06:18.560 ","End":"06:20.330","Text":"it would still be a node point,"},{"Start":"06:20.330 ","End":"06:22.260","Text":"but it would be unstable."},{"Start":"06:22.260 ","End":"06:24.650","Text":"Because we had this previously,"},{"Start":"06:24.650 ","End":"06:28.070","Text":"I just copied the sketch that we had over there."},{"Start":"06:28.070 ","End":"06:33.150","Text":"That will be it."}],"ID":10714},{"Watched":false,"Name":"exercise 7","Duration":"9m ","ChapterTopicVideoID":10350,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this exercise, we have a nonlinear planar system."},{"Start":"00:05.130 ","End":"00:08.430","Text":"This system of differential equations is"},{"Start":"00:08.430 ","End":"00:11.430","Text":"sometimes called autonomous when it\u0027s in this form,"},{"Start":"00:11.430 ","End":"00:14.400","Text":"x\u0027 is some function of x and y,"},{"Start":"00:14.400 ","End":"00:17.100","Text":"and y\u0027 is some function of x and y,"},{"Start":"00:17.100 ","End":"00:21.570","Text":"and t doesn\u0027t appear explicitly on the right-hand sides."},{"Start":"00:21.570 ","End":"00:28.530","Text":"Here we have x\u0027 is 2xy and here 2x minus y^2 plus 9 for y\u0027."},{"Start":"00:28.530 ","End":"00:29.790","Text":"In part a,"},{"Start":"00:29.790 ","End":"00:32.535","Text":"we have to find a non-constant function H(x,"},{"Start":"00:32.535 ","End":"00:42.724","Text":"y) such that every orbit or trajectory or solution of the system is the level curve"},{"Start":"00:42.724 ","End":"00:47.690","Text":"H=c for some constant c. In b we have to find"},{"Start":"00:47.690 ","End":"00:54.660","Text":"the critical points and then to classify them and see which of these are saddle points."},{"Start":"00:54.920 ","End":"00:57.785","Text":"Those are the ones we\u0027re interested in."},{"Start":"00:57.785 ","End":"01:02.015","Text":"For each value of c that gives a saddle point,"},{"Start":"01:02.015 ","End":"01:05.390","Text":"we want to sketch the level curve or level"},{"Start":"01:05.390 ","End":"01:13.680","Text":"set H=c and also throw in the critical points found from b."},{"Start":"01:13.900 ","End":"01:19.040","Text":"We start from a. I\u0027m going to start with a remark that in general,"},{"Start":"01:19.040 ","End":"01:20.495","Text":"such a system,"},{"Start":"01:20.495 ","End":"01:22.250","Text":"x\u0027 is a function of x,"},{"Start":"01:22.250 ","End":"01:25.325","Text":"y and y\u0027 is another function of x and y,"},{"Start":"01:25.325 ","End":"01:29.570","Text":"there is no general method to find such an H,"},{"Start":"01:29.570 ","End":"01:34.720","Text":"and only exists in special cases and there might be different techniques for each case."},{"Start":"01:34.720 ","End":"01:36.825","Text":"Now, in this one,"},{"Start":"01:36.825 ","End":"01:40.325","Text":"we\u0027re going to try it as an exact equation."},{"Start":"01:40.325 ","End":"01:45.755","Text":"Notice that dy by dx is dy by dt over dx by dt,"},{"Start":"01:45.755 ","End":"01:47.945","Text":"which is y\u0027 over x\u0027."},{"Start":"01:47.945 ","End":"01:53.285","Text":"The prime here refers to t. I guess t is not mentioned anywhere."},{"Start":"01:53.285 ","End":"01:56.655","Text":"I guess I should just mention it,"},{"Start":"01:56.655 ","End":"02:02.940","Text":"t is the independent variable and it\u0027s usually a physics question where t is time."},{"Start":"02:03.490 ","End":"02:10.550","Text":"From here, we can cross multiply this times this,"},{"Start":"02:10.550 ","End":"02:12.530","Text":"equals this times this,"},{"Start":"02:12.530 ","End":"02:20.795","Text":"but also bring the dy part of it to the left-hand side so it becomes minus."},{"Start":"02:20.795 ","End":"02:23.615","Text":"We get this plus this is 0,"},{"Start":"02:23.615 ","End":"02:26.330","Text":"and this is a function of x and y, call it M,"},{"Start":"02:26.330 ","End":"02:29.810","Text":"this function of x and y call it N. We know there"},{"Start":"02:29.810 ","End":"02:35.015","Text":"are necessary conditions for it to be exact."},{"Start":"02:35.015 ","End":"02:38.960","Text":"The partial derivative of M with respect to y"},{"Start":"02:38.960 ","End":"02:42.515","Text":"should equal the partial derivative of N with respect to x."},{"Start":"02:42.515 ","End":"02:49.610","Text":"This is in fact the case because for this with respect to y, it\u0027s minus 2y."},{"Start":"02:49.610 ","End":"02:51.380","Text":"Everything else here is a constant."},{"Start":"02:51.380 ","End":"02:53.120","Text":"Here with respect to x,"},{"Start":"02:53.120 ","End":"02:54.620","Text":"it\u0027s a constant times x,"},{"Start":"02:54.620 ","End":"02:57.770","Text":"so it\u0027s just that constant minus 2y."},{"Start":"02:57.770 ","End":"03:03.140","Text":"Let\u0027s look for a function H such"},{"Start":"03:03.140 ","End":"03:08.960","Text":"that the derivative of H with respect to x is this and with respect to y it\u0027s this,"},{"Start":"03:08.960 ","End":"03:12.604","Text":"which is what I just wrote here."},{"Start":"03:12.604 ","End":"03:20.995","Text":"That\u0027s not too hard to do even just by mentally integrating."},{"Start":"03:20.995 ","End":"03:26.060","Text":"Like for this one, if we know that this is a derivative with respect to x,"},{"Start":"03:26.060 ","End":"03:32.650","Text":"we can integrate it with respect to x and get x^2 minus xy^2 plus 9x."},{"Start":"03:32.650 ","End":"03:37.160","Text":"Then general, we would add a constant that is a constant function of y,"},{"Start":"03:37.160 ","End":"03:40.700","Text":"but we don\u0027t need to because this already is good enough for"},{"Start":"03:40.700 ","End":"03:44.870","Text":"this because if you differentiate this with respect to y,"},{"Start":"03:44.870 ","End":"03:50.940","Text":"you get minus 2y times dx minus 2xy, so we\u0027re good."},{"Start":"03:50.940 ","End":"03:55.125","Text":"This is our function H that was required for part a,"},{"Start":"03:55.125 ","End":"03:59.530","Text":"and now we\u0027ll move on to part b."},{"Start":"03:59.900 ","End":"04:02.700","Text":"We want the critical points."},{"Start":"04:02.700 ","End":"04:04.755","Text":"We want x\u0027,"},{"Start":"04:04.755 ","End":"04:07.770","Text":"I guess it was given as x\u0027 and y\u0027."},{"Start":"04:07.770 ","End":"04:09.930","Text":"We\u0027ve got both be 0."},{"Start":"04:09.930 ","End":"04:15.135","Text":"If we go back and look what was x\u0027 and y\u0027, there they were."},{"Start":"04:15.135 ","End":"04:16.500","Text":"This and this,"},{"Start":"04:16.500 ","End":"04:20.800","Text":"we want each of those to be 0."},{"Start":"04:21.260 ","End":"04:23.340","Text":"Look at the first 1."},{"Start":"04:23.340 ","End":"04:27.215","Text":"This 1 means that x has to be 0 or y has to be 0."},{"Start":"04:27.215 ","End":"04:29.015","Text":"I mean 1 of the 2 of them."},{"Start":"04:29.015 ","End":"04:31.445","Text":"Let\u0027s just quickly go through this,"},{"Start":"04:31.445 ","End":"04:33.350","Text":"x is 0, y is 0."},{"Start":"04:33.350 ","End":"04:35.510","Text":"Let\u0027s take the first case, y is 0."},{"Start":"04:35.510 ","End":"04:37.760","Text":"If y is 0, plug it into here."},{"Start":"04:37.760 ","End":"04:40.185","Text":"You get 2x plus 9=0,"},{"Start":"04:40.185 ","End":"04:44.225","Text":"so x is minus 4.5 and y is 0,"},{"Start":"04:44.225 ","End":"04:47.030","Text":"that gives us this critical point."},{"Start":"04:47.030 ","End":"04:49.265","Text":"Now if x is 0,"},{"Start":"04:49.265 ","End":"04:54.015","Text":"then we put x=0 in here,"},{"Start":"04:54.015 ","End":"04:58.330","Text":"you\u0027ve got minus y^2 plus 9 is 0,"},{"Start":"04:58.330 ","End":"05:04.265","Text":"so y^2=9 and y is plus or minus 3,"},{"Start":"05:04.265 ","End":"05:08.525","Text":"but x is 0, so we get these to 0 and plus or minus 3."},{"Start":"05:08.525 ","End":"05:12.650","Text":"There are 3 critical points."},{"Start":"05:12.650 ","End":"05:15.290","Text":"Next, we need the Jacobian,"},{"Start":"05:15.290 ","End":"05:20.480","Text":"and we\u0027re going to compute the Jacobian at each of the 3 critical points."},{"Start":"05:20.480 ","End":"05:22.940","Text":"This is what it is in general."},{"Start":"05:22.940 ","End":"05:25.325","Text":"As I recall, f(x,"},{"Start":"05:25.325 ","End":"05:29.780","Text":"y) was 2xy and g(x,"},{"Start":"05:29.780 ","End":"05:36.030","Text":"y) was 2x minus y^2 plus 9."},{"Start":"05:36.030 ","End":"05:44.535","Text":"Let\u0027s see. The derivatives of f with respect to x and y is 2y and 2x."},{"Start":"05:44.535 ","End":"05:46.940","Text":"This differentiates with respect to x."},{"Start":"05:46.940 ","End":"05:49.910","Text":"We\u0027ve just got 2 because this is"},{"Start":"05:49.910 ","End":"05:54.080","Text":"constant and then with respect to y minus 2y. That\u0027s the Jacobian."},{"Start":"05:54.080 ","End":"05:58.195","Text":"Now we want to plug in each of the 3 critical points."},{"Start":"05:58.195 ","End":"06:02.185","Text":"The first 1 was 4.5, 0."},{"Start":"06:02.185 ","End":"06:03.530","Text":"Plug that in here."},{"Start":"06:03.530 ","End":"06:05.570","Text":"This is the matrix you get."},{"Start":"06:05.570 ","End":"06:09.470","Text":"If you compute the eigenvalues,"},{"Start":"06:09.470 ","End":"06:13.610","Text":"you find that you get plus or minus"},{"Start":"06:13.610 ","End":"06:18.110","Text":"an imaginary part or a complex conjugate with the real part is 0."},{"Start":"06:18.110 ","End":"06:21.800","Text":"This is known to be a circle point and is in fact stable."},{"Start":"06:21.800 ","End":"06:26.159","Text":"In any event, it\u0027s not a saddle point."},{"Start":"06:26.750 ","End":"06:30.210","Text":"Let\u0027s go to the other 2."},{"Start":"06:30.210 ","End":"06:33.270","Text":"1 of them was 0 minus 3,"},{"Start":"06:33.270 ","End":"06:34.475","Text":"put that in here."},{"Start":"06:34.475 ","End":"06:36.080","Text":"This is what we get."},{"Start":"06:36.080 ","End":"06:39.485","Text":"These 2 are the eigenvalues,"},{"Start":"06:39.485 ","End":"06:43.730","Text":"2 real eigenvalues of opposite signs, saddle point."},{"Start":"06:43.730 ","End":"06:46.920","Text":"We\u0027re looking for saddle points, so this is good."},{"Start":"06:47.330 ","End":"06:53.225","Text":"The last one, 0,3 also turns out very similarly to be a saddle point."},{"Start":"06:53.225 ","End":"06:57.025","Text":"What we\u0027re interested in are these 2."},{"Start":"06:57.025 ","End":"07:02.294","Text":"Let\u0027s see which level curve is each of these on."},{"Start":"07:02.294 ","End":"07:08.930","Text":"Well, here\u0027s H. Now both of these give the same result, 0."},{"Start":"07:08.930 ","End":"07:10.835","Text":"We put x=0,"},{"Start":"07:10.835 ","End":"07:13.340","Text":"it doesn\u0027t really matter what y is,"},{"Start":"07:13.340 ","End":"07:15.875","Text":"plus or minus 3 or anything else."},{"Start":"07:15.875 ","End":"07:20.540","Text":"This thing will be 0 and that\u0027s our c. Both of"},{"Start":"07:20.540 ","End":"07:28.865","Text":"these points are on the same level curve, H=0."},{"Start":"07:28.865 ","End":"07:33.125","Text":"We can take x out of here and get this."},{"Start":"07:33.125 ","End":"07:36.230","Text":"To solve this, either x is 0,"},{"Start":"07:36.230 ","End":"07:39.055","Text":"which is the y-axis,"},{"Start":"07:39.055 ","End":"07:42.630","Text":"or x=9 minus y^2,"},{"Start":"07:42.630 ","End":"07:44.920","Text":"which is the parabola."},{"Start":"07:46.100 ","End":"07:49.475","Text":"I prepared a sketch of the parabola."},{"Start":"07:49.475 ","End":"07:52.760","Text":"Actually, there are online sites that you can feed in"},{"Start":"07:52.760 ","End":"07:58.610","Text":"the equation and it will give you the sketch."},{"Start":"07:58.610 ","End":"08:01.805","Text":"In this sketch, we just have the parabola part."},{"Start":"08:01.805 ","End":"08:05.360","Text":"We also want the y-axis."},{"Start":"08:05.360 ","End":"08:08.100","Text":"Let\u0027s try and sketch that in."},{"Start":"08:10.280 ","End":"08:17.015","Text":"Close enough. Now we want to also put in the critical points,"},{"Start":"08:17.015 ","End":"08:20.340","Text":"and those were"},{"Start":"08:25.360 ","End":"08:29.035","Text":"0, 3."},{"Start":"08:29.035 ","End":"08:34.080","Text":"Then we also have 0, minus 3."},{"Start":"08:34.080 ","End":"08:43.850","Text":"We also had minus 4.5, 0."},{"Start":"08:43.850 ","End":"08:45.860","Text":"2 saddle points,"},{"Start":"08:45.860 ","End":"08:49.495","Text":"and the other one that was a circle point."},{"Start":"08:49.495 ","End":"08:54.695","Text":"Luckily, the question didn\u0027t ask us to put arrows on or we\u0027d have to work a bit more."},{"Start":"08:54.695 ","End":"08:56.120","Text":"It didn\u0027t say so."},{"Start":"08:56.120 ","End":"08:57.560","Text":"I think this is enough."},{"Start":"08:57.560 ","End":"09:00.930","Text":"I now leave it as this, and we\u0027re done."}],"ID":10715},{"Watched":false,"Name":"exercise 8","Duration":"8m 59s","ChapterTopicVideoID":10351,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:09.210","Text":"In this exercise, we have a non-linear 2D system of differential equations."},{"Start":"00:09.210 ","End":"00:13.140","Text":"This kind is often called autonomous when"},{"Start":"00:13.140 ","End":"00:21.780","Text":"the right-hand sides just contain x and y and don\u0027t contain the independent variable,"},{"Start":"00:21.780 ","End":"00:23.730","Text":"I guess, which is T. It doesn\u0027t say,"},{"Start":"00:23.730 ","End":"00:33.780","Text":"but usually it\u0027s T. So when we have something like a function of x and y and f and g,"},{"Start":"00:33.780 ","End":"00:39.525","Text":"then that\u0027s a planar system of autonomous differential equations."},{"Start":"00:39.525 ","End":"00:42.200","Text":"For parts, first of all,"},{"Start":"00:42.200 ","End":"00:46.250","Text":"to find the stationary or critical points of the system."},{"Start":"00:46.250 ","End":"00:52.025","Text":"Then we have to find the Jacobian matrix at each of the stationary points."},{"Start":"00:52.025 ","End":"00:57.020","Text":"Then for each such stationary point to classify"},{"Start":"00:57.020 ","End":"01:03.200","Text":"it\u0027s type like is it a node or a sagittal or spiral,"},{"Start":"01:03.200 ","End":"01:05.315","Text":"whatever, and stability,"},{"Start":"01:05.315 ","End":"01:09.980","Text":"asymptotically stable, non-stable and so on."},{"Start":"01:09.980 ","End":"01:15.660","Text":"Then, to attempt a phase portrait of the system,"},{"Start":"01:15.730 ","End":"01:23.596","Text":"especially that shows its behavior near each of these stationary points and with arrows."},{"Start":"01:23.596 ","End":"01:26.645","Text":"Sometimes some questions say,"},{"Start":"01:26.645 ","End":"01:28.157","Text":"forget about the arrows here,"},{"Start":"01:28.157 ","End":"01:29.780","Text":"no we want the arrows."},{"Start":"01:29.780 ","End":"01:33.115","Text":"Let\u0027s start with a."},{"Start":"01:33.115 ","End":"01:41.450","Text":"The stationary points are where x\u0027 and y\u0027 are both 0."},{"Start":"01:41.450 ","End":"01:45.980","Text":"So we have an algebraic system to solve where"},{"Start":"01:45.980 ","End":"01:52.490","Text":"minus 3x plus y is 0 and minus 5x minus y plus 4x squared is 0."},{"Start":"01:52.490 ","End":"01:55.025","Text":"Just algebra."},{"Start":"01:55.025 ","End":"01:57.260","Text":"This is so straightforward."},{"Start":"01:57.260 ","End":"02:00.530","Text":"I\u0027ll just leave it for you if you want to check the computations."},{"Start":"02:00.530 ","End":"02:05.930","Text":"But we get the result that there are 2 stationary points, x equals 0,"},{"Start":"02:05.930 ","End":"02:09.480","Text":"y equals 0, or x equals 2,"},{"Start":"02:09.480 ","End":"02:11.010","Text":"y equals 6,"},{"Start":"02:11.010 ","End":"02:14.770","Text":"and these are for you to check if you want."},{"Start":"02:14.770 ","End":"02:17.090","Text":"Next in part b,"},{"Start":"02:17.090 ","End":"02:21.120","Text":"we want the Jacobian."},{"Start":"02:21.580 ","End":"02:31.110","Text":"The derivative of f with respect to x will be minus 3 and with respect to y it will be 1."},{"Start":"02:31.790 ","End":"02:35.460","Text":"So there\u0027s the minus 3, 1,"},{"Start":"02:35.460 ","End":"02:41.360","Text":"and as for g, with respect to x,"},{"Start":"02:41.360 ","End":"02:48.305","Text":"it will be minus 5 plus 8x or 8x minus 5,"},{"Start":"02:48.305 ","End":"02:52.370","Text":"and for y it will just be minus 1."},{"Start":"02:52.370 ","End":"02:56.315","Text":"That\u0027s the Jacobian in general."},{"Start":"02:56.315 ","End":"03:01.180","Text":"But what we want is our critical points."},{"Start":"03:01.180 ","End":"03:05.055","Text":"If we plug in the first one 0, 0."},{"Start":"03:05.055 ","End":"03:07.500","Text":"Then this 8x becomes 0."},{"Start":"03:07.500 ","End":"03:09.810","Text":"So we get minus 3,"},{"Start":"03:09.810 ","End":"03:12.615","Text":"1 minus 5, 1."},{"Start":"03:12.615 ","End":"03:16.350","Text":"The other critical point x is 2,"},{"Start":"03:16.350 ","End":"03:19.320","Text":"8 times 2 minus 5 is 11."},{"Start":"03:19.320 ","End":"03:25.440","Text":"So this is the Jacobian at the other."},{"Start":"03:25.440 ","End":"03:32.300","Text":"Now, part c is to classify these critical points."},{"Start":"03:32.300 ","End":"03:34.490","Text":"I\u0027ll start with the 0,"},{"Start":"03:34.490 ","End":"03:36.680","Text":"0 and this matrix,"},{"Start":"03:36.680 ","End":"03:38.945","Text":"we need the eigen values,"},{"Start":"03:38.945 ","End":"03:44.030","Text":"and I\u0027m going to leave you to check the calculations."},{"Start":"03:44.030 ","End":"03:48.140","Text":"The result is a complex conjugate pair,"},{"Start":"03:48.140 ","End":"03:50.970","Text":"minus 2 plus or minus 2i,"},{"Start":"03:50.970 ","End":"03:58.435","Text":"and the real part is non-zero and it\u0027s negative in fact."},{"Start":"03:58.435 ","End":"04:03.170","Text":"Under these conditions, when we have a complex conjugate with"},{"Start":"04:03.170 ","End":"04:07.870","Text":"a nonzero real part, it\u0027s asymptotically stable."},{"Start":"04:07.870 ","End":"04:10.830","Text":"This one at the critical point 2,"},{"Start":"04:10.830 ","End":"04:14.470","Text":"6 again I\u0027m going to just leave this here for you to check."},{"Start":"04:14.470 ","End":"04:17.605","Text":"The answer is that they\u0027re both real,"},{"Start":"04:17.605 ","End":"04:20.140","Text":"but they have opposite signs."},{"Start":"04:20.140 ","End":"04:23.050","Text":"I mean, square root of 20 is obviously bigger than 2."},{"Start":"04:23.050 ","End":"04:24.640","Text":"So when we take the plus, we get something"},{"Start":"04:24.640 ","End":"04:28.205","Text":"positive and the minus we get something negative."},{"Start":"04:28.205 ","End":"04:31.680","Text":"When the eigen values are real with opposite signs,"},{"Start":"04:31.680 ","End":"04:35.225","Text":"it\u0027s a subtle point and it\u0027s unstable."},{"Start":"04:35.225 ","End":"04:38.900","Text":"So that\u0027s the classification."},{"Start":"04:38.910 ","End":"04:43.240","Text":"I guess I should have really just mentioned that this is for 0,"},{"Start":"04:43.240 ","End":"04:47.470","Text":"0, and the other one is for 2, 6."},{"Start":"04:47.470 ","End":"04:50.945","Text":"Now the next part,"},{"Start":"04:50.945 ","End":"04:54.060","Text":"Part D is the sketch."},{"Start":"04:54.060 ","End":"05:00.629","Text":"I just repeated the information we have on the critical points."},{"Start":"05:00.629 ","End":"05:07.720","Text":"So I think it might help me with the saddle point would be the 2 eigen vectors."},{"Start":"05:07.720 ","End":"05:12.010","Text":"In a moment, you\u0027ll see why these can be useful for the sketch."},{"Start":"05:12.010 ","End":"05:15.795","Text":"This is the exact computation,"},{"Start":"05:15.795 ","End":"05:20.680","Text":"and in numerical terms with the plus,"},{"Start":"05:20.680 ","End":"05:25.810","Text":"it gives me this and with the minus here gives me this roughly."},{"Start":"05:25.810 ","End":"05:31.140","Text":"I found some software on the internet that calculates stuff like this,"},{"Start":"05:31.140 ","End":"05:34.520","Text":"and I\u0027ll show you an animation in a moment."},{"Start":"05:34.520 ","End":"05:39.120","Text":"This is the critical point 0, 0,"},{"Start":"05:39.120 ","End":"05:45.920","Text":"and this is the critical point 2, 6."},{"Start":"05:45.920 ","End":"05:47.030","Text":"For a saddle point,"},{"Start":"05:47.030 ","End":"05:52.310","Text":"the eigen vectors can be useful if I could draw a vector"},{"Start":"05:52.310 ","End":"06:00.605","Text":"like 0.224 across and 1 up,"},{"Start":"06:00.605 ","End":"06:02.945","Text":"not sure exactly where that would be,"},{"Start":"06:02.945 ","End":"06:04.140","Text":"something like this,"},{"Start":"06:04.140 ","End":"06:07.750","Text":"and if I drew the other vector, something like this."},{"Start":"06:07.750 ","End":"06:10.790","Text":"Then if extended the lines through this,"},{"Start":"06:10.790 ","End":"06:20.015","Text":"you\u0027d see that the arrows here go towards and the arrows on this one go away from."},{"Start":"06:20.015 ","End":"06:24.100","Text":"Anyway, it\u0027s a saddle and the other one\u0027s a spiral,"},{"Start":"06:24.100 ","End":"06:31.495","Text":"and it\u0027s asymptotically stable because we go in towards the point,"},{"Start":"06:31.495 ","End":"06:36.305","Text":"and it\u0027s also clockwise as we see."},{"Start":"06:36.305 ","End":"06:41.225","Text":"I\u0027d like to show you just a little bit about the kind of"},{"Start":"06:41.225 ","End":"06:49.010","Text":"calculators that are available online and that will be optional to see."},{"Start":"06:49.010 ","End":"06:51.560","Text":"I think we could just end it at this."},{"Start":"06:51.560 ","End":"06:56.820","Text":"So we\u0027re either done or not depending on if you want to wait for the show."},{"Start":"06:57.140 ","End":"07:04.370","Text":"So here\u0027s a computer-generated and it\u0027s the derivative of the direction field,"},{"Start":"07:04.370 ","End":"07:06.065","Text":"just arrows at first,"},{"Start":"07:06.065 ","End":"07:11.990","Text":"and now let\u0027s go to one of the critical points which was 0, 0."},{"Start":"07:11.990 ","End":"07:15.410","Text":"It was a spiral-like click somewhere,"},{"Start":"07:15.410 ","End":"07:19.205","Text":"and it gives me the orbit trajectory path."},{"Start":"07:19.205 ","End":"07:23.805","Text":"I can just click in a few places around the 0, 0,"},{"Start":"07:23.805 ","End":"07:29.540","Text":"and you see that we get the spiral and we"},{"Start":"07:29.540 ","End":"07:35.225","Text":"even know the direction of the spiral because of the arrows."},{"Start":"07:35.225 ","End":"07:37.670","Text":"First of all, it goes inwards."},{"Start":"07:37.670 ","End":"07:41.840","Text":"So it\u0027s stable, asymptotically stable."},{"Start":"07:41.840 ","End":"07:49.190","Text":"I mean, eventually it gets towards the critical point and it\u0027s also a clockwise."},{"Start":"07:49.190 ","End":"07:54.365","Text":"We can just see from the picture that the arrows here are clockwise."},{"Start":"07:54.365 ","End":"07:57.200","Text":"Now the other point was 2, 6."},{"Start":"07:57.200 ","End":"08:02.060","Text":"Here I\u0027ve got some scale 2 and where\u0027s the 6?"},{"Start":"08:02.060 ","End":"08:03.965","Text":"I think this would be it."},{"Start":"08:03.965 ","End":"08:06.755","Text":"Let me just click on here and see what happens."},{"Start":"08:06.755 ","End":"08:08.600","Text":"This is going to be our saddle point."},{"Start":"08:08.600 ","End":"08:12.230","Text":"I\u0027ll click in a few places around here,"},{"Start":"08:12.230 ","End":"08:18.880","Text":"maybe here, and here, and here."},{"Start":"08:18.880 ","End":"08:20.474","Text":"Just click a few."},{"Start":"08:20.474 ","End":"08:23.735","Text":"Each time I click, it just gives me the path."},{"Start":"08:23.735 ","End":"08:30.710","Text":"Here we see also that this really is the saddle point."},{"Start":"08:30.710 ","End":"08:36.260","Text":"At this point we\u0027re going towards and we\u0027re going towards"},{"Start":"08:36.260 ","End":"08:38.840","Text":"and on the other straight line through"},{"Start":"08:38.840 ","End":"08:43.160","Text":"the other eigenvalue we see we\u0027re getting away and away,"},{"Start":"08:43.160 ","End":"08:45.170","Text":"and you can see it\u0027s a saddle point,"},{"Start":"08:45.170 ","End":"08:48.470","Text":"and this was just for illustrative purposes,"},{"Start":"08:48.470 ","End":"08:57.455","Text":"and it\u0027s good that they have such sites with calculators for phase planes and so forth,"},{"Start":"08:57.455 ","End":"09:00.330","Text":"and I\u0027ll just leave it at that."}],"ID":10716},{"Watched":false,"Name":"exercise 9","Duration":"11m 50s","ChapterTopicVideoID":10352,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.200 ","End":"00:05.790","Text":"In this exercise, we have the following nonlinear planar,"},{"Start":"00:05.790 ","End":"00:08.400","Text":"meaning 2D system of ODEs."},{"Start":"00:08.400 ","End":"00:13.365","Text":"This system is often called autonomous."},{"Start":"00:13.365 ","End":"00:18.255","Text":"That\u0027s when the right-hand sides are functions of x and y,"},{"Start":"00:18.255 ","End":"00:23.010","Text":"and don\u0027t contain t. Also we have here"},{"Start":"00:23.010 ","End":"00:30.094","Text":"a plot of the direction vector of sample points."},{"Start":"00:30.094 ","End":"00:32.989","Text":"It\u0027s done by some software based on MATLAB."},{"Start":"00:32.989 ","End":"00:35.730","Text":"It\u0027s available on the internet."},{"Start":"00:36.970 ","End":"00:41.255","Text":"We have to identify what"},{"Start":"00:41.255 ","End":"00:47.315","Text":"the critical points are of the system not from the picture, from calculation."},{"Start":"00:47.315 ","End":"00:53.645","Text":"Then classify the type and stability of each stationary point, critical point."},{"Start":"00:53.645 ","End":"01:00.380","Text":"Then to predict the behavior as t goes to infinity of the solution,"},{"Start":"01:00.380 ","End":"01:06.160","Text":"which is also called an orbit or a trajectory passing through,"},{"Start":"01:06.160 ","End":"01:10.095","Text":"we\u0027ll see where 2.50 is,"},{"Start":"01:10.095 ","End":"01:15.330","Text":"and to sketch the solution in the phase plane."},{"Start":"01:15.410 ","End":"01:20.810","Text":"We get 4 solutions and there\u0027s 2 solutions that we"},{"Start":"01:20.810 ","End":"01:25.685","Text":"get when we let x=0 in the first equation."},{"Start":"01:25.685 ","End":"01:30.240","Text":"Then we also get another 2 solutions,"},{"Start":"01:30.470 ","End":"01:33.180","Text":"which are these 2."},{"Start":"01:33.180 ","End":"01:35.865","Text":"When we say minus plus,"},{"Start":"01:35.865 ","End":"01:37.590","Text":"it means that if this is plus,"},{"Start":"01:37.590 ","End":"01:39.890","Text":"this is minus, and when this is minus, this is plus."},{"Start":"01:39.890 ","End":"01:43.045","Text":"That\u0027s 2 more solutions."},{"Start":"01:43.045 ","End":"01:46.170","Text":"Now we have 4 of them."},{"Start":"01:46.170 ","End":"01:48.780","Text":"Now we\u0027re going to classify them."},{"Start":"01:48.780 ","End":"01:51.140","Text":"In order to classify them,"},{"Start":"01:51.140 ","End":"01:55.420","Text":"we need the eigenvalues of what?"},{"Start":"01:55.420 ","End":"01:56.630","Text":"We need to, first of all,"},{"Start":"01:56.630 ","End":"01:59.210","Text":"find the Jacobian at each of"},{"Start":"01:59.210 ","End":"02:04.325","Text":"the 4 critical points and approximate it to a linear system with constant coefficients."},{"Start":"02:04.325 ","End":"02:07.025","Text":"Now the Jacobian, I\u0027ll call it A,"},{"Start":"02:07.025 ","End":"02:09.110","Text":"it depends on x and y."},{"Start":"02:09.110 ","End":"02:11.840","Text":"We take these functions."},{"Start":"02:11.840 ","End":"02:14.750","Text":"You can quickly check if we go back,"},{"Start":"02:14.750 ","End":"02:19.175","Text":"we see here\u0027s f and g,"},{"Start":"02:19.175 ","End":"02:20.990","Text":"differentiate with respect to x and y,"},{"Start":"02:20.990 ","End":"02:23.585","Text":"differentiate with respect to x and y,"},{"Start":"02:23.585 ","End":"02:29.910","Text":"and I just wrote the result for you that the Jacobian comes out to be this quantity here."},{"Start":"02:29.910 ","End":"02:31.490","Text":"That\u0027s in general. But,"},{"Start":"02:31.490 ","End":"02:35.870","Text":"we want to now plug in each of the 4 critical points."},{"Start":"02:35.870 ","End":"02:38.810","Text":"Here we are on a new page."},{"Start":"02:38.810 ","End":"02:41.750","Text":"I\u0027ve done all the work for you,"},{"Start":"02:41.750 ","End":"02:47.390","Text":"these 4 are the 4 critical points substituted each."},{"Start":"02:47.390 ","End":"02:48.740","Text":"Let\u0027s just take 1 example,"},{"Start":"02:48.740 ","End":"02:50.960","Text":"say 0, 2,"},{"Start":"02:50.960 ","End":"02:56.415","Text":"5y plus 2x is 5 times 2 plus twice 0,"},{"Start":"02:56.415 ","End":"03:01.275","Text":"5 times 0 minus 3 times 2 is 6."},{"Start":"03:01.275 ","End":"03:04.800","Text":"Twice 2 minus 3 times 0 is 4."},{"Start":"03:04.800 ","End":"03:07.275","Text":"Similarly for the others."},{"Start":"03:07.275 ","End":"03:16.795","Text":"Now, I\u0027ve gone and computed the eigenvalues at each of these critical points."},{"Start":"03:16.795 ","End":"03:18.655","Text":"It\u0027s not that straightforward,"},{"Start":"03:18.655 ","End":"03:23.439","Text":"but if it\u0027s a triangular 2 by 2,"},{"Start":"03:23.439 ","End":"03:24.880","Text":"then we can just take these,"},{"Start":"03:24.880 ","End":"03:26.169","Text":"the main diagonal,"},{"Start":"03:26.169 ","End":"03:27.745","Text":"and these are the eigenvalues."},{"Start":"03:27.745 ","End":"03:30.155","Text":"Here we know it\u0027s 10 and 4."},{"Start":"03:30.155 ","End":"03:33.720","Text":"Here we know it\u0027s minus 10 and minus 4."},{"Start":"03:33.720 ","End":"03:36.390","Text":"But for these, we need to do the calculation,"},{"Start":"03:36.390 ","End":"03:38.535","Text":"A minus Lambda_i,"},{"Start":"03:38.535 ","End":"03:39.939","Text":"take the determinant,"},{"Start":"03:39.939 ","End":"03:44.770","Text":"and so on, and solve the quadratic in Lambda."},{"Start":"03:44.770 ","End":"03:50.883","Text":"I computed it as minus 4 plus 10,"},{"Start":"03:50.883 ","End":"03:57.830","Text":"and plus 4 minus 10 for this 1."},{"Start":"03:57.830 ","End":"04:03.335","Text":"Now, this is all we need to classify these critical points."},{"Start":"04:03.335 ","End":"04:12.960","Text":"For example, if we have 2 real same-sign numbers and we know it,"},{"Start":"04:12.960 ","End":"04:18.755","Text":"say node, and if they\u0027re both positive, it\u0027s unstable."},{"Start":"04:18.755 ","End":"04:21.245","Text":"But if they\u0027re both negative,"},{"Start":"04:21.245 ","End":"04:25.709","Text":"then it asymptotically stable."},{"Start":"04:27.520 ","End":"04:32.690","Text":"Now when we have 2 eigenvalues of opposite signs,"},{"Start":"04:32.690 ","End":"04:34.218","Text":"it\u0027s always a saddle point."},{"Start":"04:34.218 ","End":"04:36.423","Text":"And a saddle point is unstable,"},{"Start":"04:36.423 ","End":"04:37.960","Text":"so here and here,"},{"Start":"04:37.960 ","End":"04:39.830","Text":"we have unstable saddle points."},{"Start":"04:39.830 ","End":"04:43.960","Text":"That\u0027s the classification for the 4 critical points."},{"Start":"04:43.960 ","End":"04:50.530","Text":"Now we want to get to the part with the sketch."},{"Start":"04:52.310 ","End":"04:57.875","Text":"I will show you the sketch first and then I\u0027ll explain stuff about it."},{"Start":"04:57.875 ","End":"05:00.890","Text":"The critical points, well, there\u0027s 2 we can\u0027t see,"},{"Start":"05:00.890 ","End":"05:03.830","Text":"there was 0, 2 and 0 minus 2,"},{"Start":"05:03.830 ","End":"05:07.800","Text":"which would be this point here,"},{"Start":"05:07.800 ","End":"05:13.090","Text":"and minus 2 and 0, that\u0027s this here."},{"Start":"05:13.090 ","End":"05:18.490","Text":"The other ones are minus 2 and 1/2, 1 and 1/2."},{"Start":"05:18.490 ","End":"05:26.890","Text":"Minus 2 and 1/2 plus 1/2 would be here."},{"Start":"05:26.990 ","End":"05:36.730","Text":"The other 1 plus 2 and 1/2 minus 1/2 would be here."},{"Start":"05:38.120 ","End":"05:44.490","Text":"It makes sense that these 2 are nodes,"},{"Start":"05:44.490 ","End":"05:49.210","Text":"these lines leading out of them going away from this point,"},{"Start":"05:49.210 ","End":"05:51.100","Text":"which is why it\u0027s unstable,"},{"Start":"05:51.100 ","End":"05:52.600","Text":"and here all the arrows,"},{"Start":"05:52.600 ","End":"05:56.601","Text":"all the orbits lead towards the point at infinity,"},{"Start":"05:56.601 ","End":"05:58.795","Text":"that\u0027s why it\u0027s asymptotically stable."},{"Start":"05:58.795 ","End":"06:02.420","Text":"These 2 are clearly saddle points."},{"Start":"06:02.940 ","End":"06:08.890","Text":"I guess if you took lines parallel to eigenvalues,"},{"Start":"06:08.890 ","End":"06:11.750","Text":"you\u0027d get a cross here."},{"Start":"06:11.790 ","End":"06:16.650","Text":"On 1 diagonal or line,"},{"Start":"06:16.650 ","End":"06:18.110","Text":"you\u0027d get them going away,"},{"Start":"06:18.110 ","End":"06:20.135","Text":"and on the other 1 going towards."},{"Start":"06:20.135 ","End":"06:23.640","Text":"Here, also you get towards going away."},{"Start":"06:25.270 ","End":"06:30.000","Text":"Similarly, the other 1, these are the saddle points."},{"Start":"06:30.040 ","End":"06:33.740","Text":"I\u0027m also going to mark not a critical point,"},{"Start":"06:33.740 ","End":"06:39.410","Text":"but the point we were asked about to investigate was 2 and 1/2, 0."},{"Start":"06:39.410 ","End":"06:44.945","Text":"Over here, we have to answer questions about it."},{"Start":"06:44.945 ","End":"06:50.315","Text":"By the way, at the end, I\u0027m going to show you an animation of how we get this."},{"Start":"06:50.315 ","End":"06:53.930","Text":"There are now sites on the Internet that will do"},{"Start":"06:53.930 ","End":"07:00.260","Text":"these computations and will give you as many orbital trajectories as you like,"},{"Start":"07:00.260 ","End":"07:02.450","Text":"just by clicking on a point,"},{"Start":"07:02.450 ","End":"07:03.699","Text":"it will give you the trajectory."},{"Start":"07:03.699 ","End":"07:05.645","Text":"That\u0027s at the end if you\u0027re interested."},{"Start":"07:05.645 ","End":"07:10.700","Text":"We were asked to make a guess about what happens when"},{"Start":"07:10.700 ","End":"07:16.280","Text":"t goes to infinity around this point 2, and 1/2, 0."},{"Start":"07:16.280 ","End":"07:20.675","Text":"I just highlighted the trajectory,"},{"Start":"07:20.675 ","End":"07:22.220","Text":"make it even darker."},{"Start":"07:22.220 ","End":"07:28.320","Text":"That goes through here and goes to infinity."},{"Start":"07:28.320 ","End":"07:33.920","Text":"It looks like it reaches this line here,"},{"Start":"07:33.920 ","End":"07:38.150","Text":"which is the line through the critical point that\u0027s"},{"Start":"07:38.150 ","End":"07:42.220","Text":"parallel to the eigenvector through that point."},{"Start":"07:42.220 ","End":"07:44.370","Text":"We\u0027ll need to compute that."},{"Start":"07:44.370 ","End":"07:47.640","Text":"I just drew the line and you\u0027ll know which 1 I mean."},{"Start":"07:47.640 ","End":"07:49.790","Text":"It looks like the trajectory, the orbit,"},{"Start":"07:49.790 ","End":"07:53.075","Text":"goes towards the line."},{"Start":"07:53.075 ","End":"07:54.469","Text":"This would be an asymptote."},{"Start":"07:54.469 ","End":"07:55.970","Text":"Let\u0027s do some computation."},{"Start":"07:55.970 ","End":"08:04.030","Text":"Now, the eigenvalues at this point here are this and this."},{"Start":"08:04.030 ","End":"08:06.125","Text":"I can\u0027t say I computed it myself."},{"Start":"08:06.125 ","End":"08:12.275","Text":"There are even websites that have calculators that calculate the eigenvalues for you."},{"Start":"08:12.275 ","End":"08:14.560","Text":"Well, 1 way or another,"},{"Start":"08:14.560 ","End":"08:19.460","Text":"these 2, actually this was given as 1 here,"},{"Start":"08:19.460 ","End":"08:20.960","Text":"an 8 and a 1/3 here,"},{"Start":"08:20.960 ","End":"08:25.180","Text":"but I multiplied by 3,"},{"Start":"08:25.180 ","End":"08:26.855","Text":"so it wouldn\u0027t get fractions."},{"Start":"08:26.855 ","End":"08:31.045","Text":"Anyway, this is in the ratio 25 across 3 up,"},{"Start":"08:31.045 ","End":"08:34.310","Text":"and this is the 1 that is this asymptote."},{"Start":"08:34.310 ","End":"08:36.935","Text":"There is another asymptote that goes, I guess,"},{"Start":"08:36.935 ","End":"08:41.670","Text":"45 degrees, 1 minus 1 would be parallel to it."},{"Start":"08:43.930 ","End":"08:46.760","Text":"That\u0027s what I drew in this green."},{"Start":"08:46.760 ","End":"08:49.264","Text":"If you want the equation of the asymptote,"},{"Start":"08:49.264 ","End":"08:50.855","Text":"here it is,"},{"Start":"08:50.855 ","End":"08:52.820","Text":"y minus the y of the point,"},{"Start":"08:52.820 ","End":"08:56.165","Text":"which is y plus 1/2 equals slope."},{"Start":"08:56.165 ","End":"08:59.210","Text":"The slope would be 3 over 25,"},{"Start":"08:59.210 ","End":"09:03.000","Text":"the y over the x, rise over run."},{"Start":"09:03.160 ","End":"09:06.830","Text":"Well, that\u0027s really it,"},{"Start":"09:06.830 ","End":"09:10.392","Text":"except for the part about the animation which is optional,"},{"Start":"09:10.392 ","End":"09:12.440","Text":"so you can stay and watch."},{"Start":"09:12.440 ","End":"09:17.000","Text":"Here we are with the optional graphical part."},{"Start":"09:17.000 ","End":"09:19.250","Text":"We had the critical points."},{"Start":"09:19.250 ","End":"09:21.140","Text":"One of them was 0,2,"},{"Start":"09:21.140 ","End":"09:25.050","Text":"that\u0027s here, I\u0027m going to click there."},{"Start":"09:25.150 ","End":"09:27.890","Text":"Depending where I click,"},{"Start":"09:27.890 ","End":"09:32.450","Text":"I get different trajectories."},{"Start":"09:32.450 ","End":"09:36.790","Text":"But, you can see that the arrows are all going away."},{"Start":"09:36.790 ","End":"09:41.870","Text":"This is the unstable node."},{"Start":"09:41.870 ","End":"09:46.550","Text":"Then we already can see the other 1,"},{"Start":"09:46.550 ","End":"09:48.350","Text":"that was 0 minus 2,"},{"Start":"09:48.350 ","End":"09:50.705","Text":"that\u0027s this point here."},{"Start":"09:50.705 ","End":"09:54.478","Text":"You can click nearby it,"},{"Start":"09:54.478 ","End":"10:01.440","Text":"and get all kinds of orbits, trajectories."},{"Start":"10:01.440 ","End":"10:08.450","Text":"This time, they lead into the critical point."},{"Start":"10:08.450 ","End":"10:11.929","Text":"This is a asymptotically stable node."},{"Start":"10:11.929 ","End":"10:19.450","Text":"Now we also had minus 2 and 1/2, 1 and 1/2."},{"Start":"10:19.450 ","End":"10:24.590","Text":"This one was the critical point and see what happens if I click there or here,"},{"Start":"10:24.590 ","End":"10:27.740","Text":"or here, or here."},{"Start":"10:27.740 ","End":"10:31.070","Text":"I think we can see that this is a saddle point."},{"Start":"10:31.070 ","End":"10:33.445","Text":"It shows."},{"Start":"10:33.445 ","End":"10:40.215","Text":"There are arrows coming in and going away."},{"Start":"10:40.215 ","End":"10:43.385","Text":"There are asymptotes if we draw it more carefully."},{"Start":"10:43.385 ","End":"10:45.810","Text":"Similarly, let\u0027s see,"},{"Start":"10:45.810 ","End":"10:48.210","Text":"2 and 1/2 is here,"},{"Start":"10:48.210 ","End":"10:51.270","Text":"minus a 1/2, this 1."},{"Start":"10:51.270 ","End":"10:53.715","Text":"That\u0027s also, I\u0027m clicking,"},{"Start":"10:53.715 ","End":"10:58.635","Text":"click, nice toy,"},{"Start":"10:58.635 ","End":"11:00.900","Text":"nice tool for drawing."},{"Start":"11:00.900 ","End":"11:02.520","Text":"Another saddle point,"},{"Start":"11:02.520 ","End":"11:06.455","Text":"arrows going in and arrows going out, unstable."},{"Start":"11:06.455 ","End":"11:10.330","Text":"Then we had that to investigate."},{"Start":"11:10.330 ","End":"11:13.855","Text":"What was it? 2 and 1/2, 0,"},{"Start":"11:13.855 ","End":"11:18.260","Text":"that would be the trajectory through this point."},{"Start":"11:18.260 ","End":"11:20.545","Text":"Well, here we see it."},{"Start":"11:20.545 ","End":"11:22.230","Text":"We see the trajectory,"},{"Start":"11:22.230 ","End":"11:25.850","Text":"it\u0027s the upper one of these 2 that are close together."},{"Start":"11:25.850 ","End":"11:33.150","Text":"You can see that this goes towards this asymptote."},{"Start":"11:33.310 ","End":"11:40.025","Text":"The solution trajectory orbit through this point goes"},{"Start":"11:40.025 ","End":"11:47.210","Text":"according to this arrow and infinity towards the asymptote through this saddle point."},{"Start":"11:47.210 ","End":"11:51.630","Text":"We\u0027re done with the optional animated part."}],"ID":10717},{"Watched":false,"Name":"exercise 10","Duration":"8m 17s","ChapterTopicVideoID":10353,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.130","Text":"In this exercise, we have a nonlinear system."},{"Start":"00:05.130 ","End":"00:07.980","Text":"It\u0027s planar or it\u0027s in 2D."},{"Start":"00:07.980 ","End":"00:13.710","Text":"We call this system autonomous when"},{"Start":"00:13.710 ","End":"00:18.630","Text":"the right-hand sides are functions of just x and y and don\u0027t"},{"Start":"00:18.630 ","End":"00:21.705","Text":"involve the independent variable which is usually"},{"Start":"00:21.705 ","End":"00:29.800","Text":"t. There\u0027s no general technique."},{"Start":"00:29.990 ","End":"00:32.010","Text":"In this exercise,"},{"Start":"00:32.010 ","End":"00:35.940","Text":"we have a nonlinear system as follows."},{"Start":"00:35.940 ","End":"00:38.360","Text":"It\u0027s planar or it\u0027s in 2D."},{"Start":"00:38.360 ","End":"00:41.240","Text":"It\u0027s called autonomous when"},{"Start":"00:41.240 ","End":"00:47.100","Text":"the right-hand sides here don\u0027t involve the independent variable."},{"Start":"00:47.410 ","End":"00:50.120","Text":"They are just functions of x and y."},{"Start":"00:50.120 ","End":"00:51.920","Text":"The independent variable, doesn\u0027t say,"},{"Start":"00:51.920 ","End":"00:55.280","Text":"it\u0027s probably t. In part A,"},{"Start":"00:55.280 ","End":"00:57.080","Text":"we have to find the critical points,"},{"Start":"00:57.080 ","End":"01:00.040","Text":"so stationary points of the system."},{"Start":"01:00.040 ","End":"01:06.530","Text":"Then in B, we write down the linearization of the system at each critical point where we"},{"Start":"01:06.530 ","End":"01:15.470","Text":"approximate a system with constant coefficients that\u0027s linear at each critical point."},{"Start":"01:15.470 ","End":"01:20.780","Text":"Then we want to classify the type and stability of each critical point."},{"Start":"01:20.780 ","End":"01:26.930","Text":"Finally, to sketch a phase portrait with arrows."},{"Start":"01:26.930 ","End":"01:29.885","Text":"Sometimes it\u0027s with arrows and sometimes it\u0027s without arrows."},{"Start":"01:29.885 ","End":"01:34.560","Text":"The orbits are also called trajectories."},{"Start":"01:35.320 ","End":"01:37.955","Text":"Now the critical points,"},{"Start":"01:37.955 ","End":"01:39.930","Text":"stationary points same thing,"},{"Start":"01:39.930 ","End":"01:43.070","Text":"occur when x\u0027 and y\u0027 are both 0,"},{"Start":"01:43.070 ","End":"01:45.995","Text":"when the derivative is 0,"},{"Start":"01:45.995 ","End":"01:49.685","Text":"which means that each of these functions is 0."},{"Start":"01:49.685 ","End":"01:53.975","Text":"We get x^2 minus 9 is 0 and 2xy is 0."},{"Start":"01:53.975 ","End":"01:57.295","Text":"Two equations and 2 unknowns."},{"Start":"01:57.295 ","End":"02:01.500","Text":"Now, the solutions aren\u0027t just these."},{"Start":"02:01.500 ","End":"02:02.520","Text":"I just wrote them."},{"Start":"02:02.520 ","End":"02:05.685","Text":"It\u0027s a simple algebra but I\u0027ll say a few words."},{"Start":"02:05.685 ","End":"02:06.930","Text":"From the first equation,"},{"Start":"02:06.930 ","End":"02:08.070","Text":"it just has x in it,"},{"Start":"02:08.070 ","End":"02:11.220","Text":"x^2 is 9 so x is plus or minus 3."},{"Start":"02:11.220 ","End":"02:13.530","Text":"Certainly, x is not 0,"},{"Start":"02:13.530 ","End":"02:14.795","Text":"it\u0027s plus or minus 3."},{"Start":"02:14.795 ","End":"02:20.845","Text":"If x is not 0, we can divide by x and we can divide by 2 and we get that y is 0."},{"Start":"02:20.845 ","End":"02:26.030","Text":"So y is 0 either way and we have x is either 3 or minus 3."},{"Start":"02:26.120 ","End":"02:29.205","Text":"That\u0027s that."},{"Start":"02:29.205 ","End":"02:32.070","Text":"Moving on to part B where we have"},{"Start":"02:32.070 ","End":"02:40.715","Text":"to write the linearization at each of the critical points,"},{"Start":"02:40.715 ","End":"02:43.400","Text":"we need the Jacobian for that."},{"Start":"02:43.400 ","End":"02:51.125","Text":"The Jacobian is this matrix of partial derivatives of f and g with respect to x and y."},{"Start":"02:51.125 ","End":"02:55.370","Text":"If you differentiate this with respect to x and y, you get this."},{"Start":"02:55.370 ","End":"03:00.320","Text":"This is our Jacobian matrix."},{"Start":"03:00.320 ","End":"03:06.570","Text":"Now, we take this Jacobian and substitute each critical point."},{"Start":"03:06.950 ","End":"03:13.055","Text":"Now, we plug in each of these into the Jacobian, the 3, 0."},{"Start":"03:13.055 ","End":"03:17.635","Text":"Let\u0027s do that one first and then you see you just get the 2x,"},{"Start":"03:17.635 ","End":"03:20.715","Text":"6, 6 because y is 0."},{"Start":"03:20.715 ","End":"03:25.500","Text":"Similarly here, we just get minus 6 instead of 6."},{"Start":"03:25.500 ","End":"03:33.210","Text":"Now, we want the linearization which means"},{"Start":"03:33.210 ","End":"03:42.055","Text":"treating the Jacobian A like a constant and writing x\u0027 equals Ax like we were used to."},{"Start":"03:42.055 ","End":"03:48.320","Text":"This is an approximation for the differential equation system around"},{"Start":"03:48.320 ","End":"03:55.080","Text":"each critical point in some neighborhood of 3, 0 nearby."},{"Start":"03:55.080 ","End":"03:56.820","Text":"It\u0027s roughly this."},{"Start":"03:56.820 ","End":"04:02.185","Text":"These are the 2 linearizations and we know how to handle those,"},{"Start":"04:02.185 ","End":"04:04.520","Text":"how to classify them."},{"Start":"04:04.520 ","End":"04:08.690","Text":"We just find the eigenvalues."},{"Start":"04:08.690 ","End":"04:11.570","Text":"Now, when it\u0027s a diagonal matrix,"},{"Start":"04:11.570 ","End":"04:15.005","Text":"the eigenvalues are just the entries on the diagonal."},{"Start":"04:15.005 ","End":"04:20.870","Text":"Here we have just one eigenvalue or you could say it\u0027s"},{"Start":"04:20.870 ","End":"04:29.375","Text":"a double repeated eigenvalue 6 and 6 and then the other one is minus 6 and minus 6."},{"Start":"04:29.375 ","End":"04:32.045","Text":"When we have this case,"},{"Start":"04:32.045 ","End":"04:38.570","Text":"then it\u0027s a proper node with the difference being that if that repeated eigenvalue is"},{"Start":"04:38.570 ","End":"04:47.565","Text":"positive then it\u0027s unstable and if it\u0027s negative then it\u0027s asymptotically stable."},{"Start":"04:47.565 ","End":"04:51.905","Text":"There\u0027s also another name for a proper node,"},{"Start":"04:51.905 ","End":"04:54.995","Text":"it\u0027s also called a star point."},{"Start":"04:54.995 ","End":"04:58.770","Text":"We\u0027ll see when we do the sketch."},{"Start":"04:59.750 ","End":"05:03.585","Text":"Now, we come to the sketch part."},{"Start":"05:03.585 ","End":"05:07.530","Text":"Usually, you would do this with something"},{"Start":"05:07.530 ","End":"05:12.380","Text":"computer-aided but if you started doing it by hand,"},{"Start":"05:12.380 ","End":"05:20.890","Text":"you might start by drawing a pair of axes like a y-axis and an x-axis."},{"Start":"05:20.890 ","End":"05:23.445","Text":"Just to label them,"},{"Start":"05:23.445 ","End":"05:26.235","Text":"that would be the x, that would be the y."},{"Start":"05:26.235 ","End":"05:31.910","Text":"Then we want to put our critical points in, 3,"},{"Start":"05:31.910 ","End":"05:36.570","Text":"0 would be here and minus 3,"},{"Start":"05:36.570 ","End":"05:38.985","Text":"0 would be here."},{"Start":"05:38.985 ","End":"05:43.155","Text":"This one is the unstable star node."},{"Start":"05:43.155 ","End":"05:47.000","Text":"You might draw a few lines coming out of it."},{"Start":"05:47.000 ","End":"05:56.010","Text":"They are all not doing a very good job here but"},{"Start":"05:56.010 ","End":"06:06.129","Text":"these all have the arrows going away from the node."},{"Start":"06:06.470 ","End":"06:14.960","Text":"Then similarly, this one is the asymptotically stable star node."},{"Start":"06:14.960 ","End":"06:18.110","Text":"I\u0027ll draw a few of these."},{"Start":"06:18.110 ","End":"06:27.270","Text":"This time the arrows would go inwards and so on."},{"Start":"06:27.470 ","End":"06:32.990","Text":"Then you might figure that somehow these will join up so"},{"Start":"06:32.990 ","End":"06:38.675","Text":"you might start drawing lines here."},{"Start":"06:38.675 ","End":"06:39.920","Text":"It looks a mess."},{"Start":"06:39.920 ","End":"06:43.085","Text":"Let me show you how it should look."},{"Start":"06:43.085 ","End":"06:45.980","Text":"Wait, I\u0027ll just add a few more let\u0027s say."},{"Start":"06:45.980 ","End":"06:48.900","Text":"This looks terrible."},{"Start":"06:49.820 ","End":"06:55.050","Text":"I might go like this."},{"Start":"06:55.910 ","End":"07:02.810","Text":"That\u0027s why we would usually start off with something like MATLAB"},{"Start":"07:02.810 ","End":"07:06.720","Text":"assisted to get all the directions at each point"},{"Start":"07:06.720 ","End":"07:11.495","Text":"which you could compute by substituting in the f(x y), g(x, y)."},{"Start":"07:11.495 ","End":"07:17.460","Text":"Then I guess this would be the minus 3,"},{"Start":"07:17.460 ","End":"07:19.635","Text":"0, this would be 3, 0."},{"Start":"07:19.635 ","End":"07:23.685","Text":"Then you can trace these and get"},{"Start":"07:23.685 ","End":"07:31.560","Text":"lines here and maybe here."},{"Start":"07:31.560 ","End":"07:37.500","Text":"Let me just show you something that looks a bit better."},{"Start":"07:38.120 ","End":"07:40.530","Text":"Here we are."},{"Start":"07:40.530 ","End":"07:42.675","Text":"Again, computer-aided."},{"Start":"07:42.675 ","End":"07:46.740","Text":"By hand is not the way to go but you do see that we have"},{"Start":"07:46.740 ","End":"07:52.860","Text":"these 2 star nodes or proper nodes."},{"Start":"07:52.860 ","End":"07:55.760","Text":"This one where the arrows go inwards is the"},{"Start":"07:55.760 ","End":"07:59.730","Text":"asymptotically stable and this one is the unstable."},{"Start":"07:59.950 ","End":"08:03.200","Text":"But when you start getting away from these points,"},{"Start":"08:03.200 ","End":"08:04.730","Text":"it\u0027s no longer the star shape."},{"Start":"08:04.730 ","End":"08:12.870","Text":"It starts to bend and you draw maybe more than a few of these orbits or trajectories."},{"Start":"08:12.870 ","End":"08:15.585","Text":"This is the idea. This is what it roughly looks like."},{"Start":"08:15.585 ","End":"08:18.310","Text":"I think we\u0027ll leave it at that."}],"ID":10718},{"Watched":false,"Name":"exercise 11","Duration":"7m 34s","ChapterTopicVideoID":10354,"CourseChapterTopicPlaylistID":16411,"HasSubtitles":true,"ThumbnailPath":null,"UploadDate":null,"DurationForVideoObject":null,"Description":null,"MetaTitle":null,"MetaDescription":null,"Canonical":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.325","Text":"In this exercise, we have a system of differential equations,"},{"Start":"00:05.325 ","End":"00:09.030","Text":"homogeneous linear constant coefficients planar,"},{"Start":"00:09.030 ","End":"00:14.565","Text":"some 2D as follows where x is x_1, x_2."},{"Start":"00:14.565 ","End":"00:19.390","Text":"Notice that the determinant of this matrix is not 0,"},{"Start":"00:19.390 ","End":"00:24.644","Text":"so we know that this means that there\u0027s only 1 critical point and it\u0027s at the origin."},{"Start":"00:24.644 ","End":"00:28.080","Text":"Our task is to classify the critical point,"},{"Start":"00:28.080 ","End":"00:32.700","Text":"which means what type is it and what is its stability?"},{"Start":"00:32.700 ","End":"00:39.020","Text":"I\u0027m going to do the technical part quickly because we\u0027re used to it,"},{"Start":"00:39.020 ","End":"00:42.980","Text":"and spend more time on the sketch of the phase portrait."},{"Start":"00:42.980 ","End":"00:46.715","Text":"Some of the technical details I\u0027m omitting,"},{"Start":"00:46.715 ","End":"00:50.840","Text":"like finding the eigenvalues of a matrix."},{"Start":"00:50.840 ","End":"00:53.975","Text":"If you compute it,"},{"Start":"00:53.975 ","End":"01:00.435","Text":"if you subtract Lambda i from this or other way round,"},{"Start":"01:00.435 ","End":"01:02.705","Text":"then you\u0027ll get an equation."},{"Start":"01:02.705 ","End":"01:07.280","Text":"I remember the characteristic equation came out to"},{"Start":"01:07.280 ","End":"01:13.890","Text":"be Lambda^2 minus 10 Lambda plus 25=0, and that\u0027s Lambda minus 5^2."},{"Start":"01:14.020 ","End":"01:20.295","Text":"We see we have a double repeated eigenvalue,"},{"Start":"01:20.295 ","End":"01:24.215","Text":"there is 2 of them, they\u0027re both 5 or you can say just 1 of them."},{"Start":"01:24.215 ","End":"01:25.820","Text":"In this case,"},{"Start":"01:25.820 ","End":"01:31.290","Text":"we have to do further investigating because it could be 1 or 2 eigenvectors,"},{"Start":"01:31.290 ","End":"01:33.615","Text":"I won\u0027t keep you in suspense."},{"Start":"01:33.615 ","End":"01:39.530","Text":"Turns out that there is only 1 eigenvector and this is it,"},{"Start":"01:39.530 ","End":"01:40.760","Text":"or at least it\u0027s a basis this,"},{"Start":"01:40.760 ","End":"01:45.020","Text":"and of course any non-zero multiple of this is also an eigenvector."},{"Start":"01:45.020 ","End":"01:47.720","Text":"When we have only 1 eigenvector,"},{"Start":"01:47.720 ","End":"01:51.990","Text":"it means that we have an improper node."},{"Start":"01:52.580 ","End":"01:57.740","Text":"There\u0027s 2 kinds of improper node."},{"Start":"01:57.740 ","End":"02:01.820","Text":"It could be unstable or asymptotically stable depending on"},{"Start":"02:01.820 ","End":"02:06.390","Text":"whether this common eigenvalue is positive or negative."},{"Start":"02:06.390 ","End":"02:12.343","Text":"When it\u0027s positive, then the improper node is unstable."},{"Start":"02:12.343 ","End":"02:16.640","Text":"Now, we come to the sketch part."},{"Start":"02:16.640 ","End":"02:21.980","Text":"There\u0027s more than 1 way to do a sketch depending on whether you want"},{"Start":"02:21.980 ","End":"02:28.520","Text":"to do it with MATLAB or help of a computer."},{"Start":"02:28.520 ","End":"02:33.410","Text":"Let me start with one way as just to draw some axes,"},{"Start":"02:33.410 ","End":"02:34.640","Text":"and then on the axes,"},{"Start":"02:34.640 ","End":"02:37.940","Text":"you would put the eigenvector."},{"Start":"02:37.940 ","End":"02:40.325","Text":"The eigenvector, if I remember,"},{"Start":"02:40.325 ","End":"02:46.350","Text":"was the vector 1 minus 2, I believe."},{"Start":"02:46.350 ","End":"02:48.390","Text":"Then we could mark,"},{"Start":"02:48.390 ","End":"02:52.210","Text":"let\u0027s say 1 minus 2 might be here."},{"Start":"02:52.210 ","End":"02:59.030","Text":"Then you would also put maybe the origin as the critical point,"},{"Start":"02:59.030 ","End":"03:04.970","Text":"and then its negative would be minus 1,2 would be somewhere here,"},{"Start":"03:04.970 ","End":"03:07.365","Text":"and then draw the line through them."},{"Start":"03:07.365 ","End":"03:14.890","Text":"Because it\u0027s unstable, the arrows are going away from the origin."},{"Start":"03:14.890 ","End":"03:18.440","Text":"Then when it\u0027s an improper node,"},{"Start":"03:18.440 ","End":"03:21.890","Text":"we know that all the orbits,"},{"Start":"03:21.890 ","End":"03:28.430","Text":"they start out from the origin parallel to the asymptote."},{"Start":"03:28.430 ","End":"03:29.705","Text":"This would be the asymptote,"},{"Start":"03:29.705 ","End":"03:31.250","Text":"but there\u0027s 2 possibilities,"},{"Start":"03:31.250 ","End":"03:35.005","Text":"they could go here and eventually they double"},{"Start":"03:35.005 ","End":"03:39.905","Text":"back and at infinity it becomes parallel to this again."},{"Start":"03:39.905 ","End":"03:43.950","Text":"But it could also start on the other side,"},{"Start":"03:43.950 ","End":"03:48.495","Text":"it goes something like this so we have to know which it is."},{"Start":"03:48.495 ","End":"03:57.120","Text":"One way is just to substitute a point, not to scale,"},{"Start":"03:57.120 ","End":"04:01.945","Text":"but let\u0027s say 1,0,"},{"Start":"04:01.945 ","End":"04:07.000","Text":"I would have to go back and see where we are."},{"Start":"04:07.970 ","End":"04:10.860","Text":"If it\u0027s 1,0,"},{"Start":"04:10.860 ","End":"04:17.895","Text":"then we plug in 1,0 for x then we get,"},{"Start":"04:17.895 ","End":"04:20.145","Text":"just write that 9,"},{"Start":"04:20.145 ","End":"04:21.555","Text":"2, minus 8,"},{"Start":"04:21.555 ","End":"04:24.798","Text":"1 times 1,0,"},{"Start":"04:24.798 ","End":"04:30.330","Text":"this would give us 9,"},{"Start":"04:30.330 ","End":"04:34.005","Text":"and this would give us minus 8,"},{"Start":"04:34.005 ","End":"04:41.480","Text":"it\u0027s going this way and so that means that this case is ruled out."},{"Start":"04:41.480 ","End":"04:46.065","Text":"Then we know the general shape will be,"},{"Start":"04:46.065 ","End":"04:49.370","Text":"well, they don\u0027t cut each other but it\u0027s starts off,"},{"Start":"04:49.370 ","End":"04:53.310","Text":"it\u0027ll go here this way parallel with"},{"Start":"04:53.310 ","End":"05:03.185","Text":"the asymptote in this direction and then ending up at infinity also parallel."},{"Start":"05:03.185 ","End":"05:09.810","Text":"Over here, it also goes on this side,"},{"Start":"05:09.810 ","End":"05:13.160","Text":"it\u0027s always the symmetric image of it so we"},{"Start":"05:13.160 ","End":"05:20.225","Text":"have arrows like this."},{"Start":"05:20.225 ","End":"05:24.170","Text":"You have a general idea that this is what it looks like,"},{"Start":"05:24.170 ","End":"05:27.695","Text":"this improper node that goes in this direction."},{"Start":"05:27.695 ","End":"05:35.810","Text":"We could also get a computerized sketch of the vectors at each point."},{"Start":"05:35.810 ","End":"05:38.495","Text":"There are websites that do this."},{"Start":"05:38.495 ","End":"05:42.980","Text":"Here\u0027s 1, let\u0027s mark the origin as"},{"Start":"05:42.980 ","End":"05:49.243","Text":"the critical point and also the eigenvalue,"},{"Start":"05:49.243 ","End":"05:50.405","Text":"the line through it."},{"Start":"05:50.405 ","End":"05:53.555","Text":"The eigenvalue was 1,"},{"Start":"05:53.555 ","End":"05:57.515","Text":"minus 2 was here."},{"Start":"05:57.515 ","End":"06:01.600","Text":"Here\u0027s minus 1,2,"},{"Start":"06:01.600 ","End":"06:09.880","Text":"let\u0027s see if I can freehand it, not so great."},{"Start":"06:09.920 ","End":"06:12.310","Text":"That\u0027s a bit better,"},{"Start":"06:12.310 ","End":"06:16.850","Text":"and we can see the arrows going this way,"},{"Start":"06:18.350 ","End":"06:22.750","Text":"and over here they\u0027re going this way."},{"Start":"06:24.450 ","End":"06:28.600","Text":"We can try and track some of these,"},{"Start":"06:28.600 ","End":"06:32.425","Text":"go along with the arrows."},{"Start":"06:32.425 ","End":"06:39.228","Text":"You can see that this one goes and it will end up here and here, there they go."},{"Start":"06:39.228 ","End":"06:41.365","Text":"The other way, like we said,"},{"Start":"06:41.365 ","End":"06:46.645","Text":"I\u0027m just doing this quick just to get the idea,"},{"Start":"06:46.645 ","End":"06:54.370","Text":"they all start off parallel asymptotic"},{"Start":"06:54.370 ","End":"07:03.080","Text":"to this line through the eigenvector and they end up parallel at infinity."},{"Start":"07:03.080 ","End":"07:04.850","Text":"It ends up being parallel,"},{"Start":"07:04.850 ","End":"07:07.770","Text":"so lets put some arrows on these."},{"Start":"07:09.530 ","End":"07:18.480","Text":"Why don\u0027t I show you a couple of better pictures than these?"},{"Start":"07:20.330 ","End":"07:26.915","Text":"Here they are just sketched a bit better, more satisfying."},{"Start":"07:26.915 ","End":"07:29.930","Text":"Anyway, you\u0027ve got the idea."},{"Start":"07:29.930 ","End":"07:34.440","Text":"Then let\u0027s say we\u0027re done."}],"ID":10719}],"Thumbnail":null,"ID":16411}]

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1.1

1