 fluid. So you must finish at what time? 16. You want to finish at what time? Okay, okay, perfect. So we are very happy to have the second lecture. So I advise you, I advise you to take, because this lecture is very pedagogical, so it's good to, it's kind of course. So I advise you to take a look at, to take a look. Welcome back. Thank you. After the introduction part yesterday, you heard from different points of view, as I noted, Jonah also introduced iPhone 6 hands. So you've heard from me, I've told you a couple of weeks ago, what you've done in collaboration, how you performed in your own form yesterday, okay. And so we actually had a lot of fun showing you from back then already. Story, we can maybe split it into two parts in time also, I think it's a better for you actually to hear it, really repeating ourselves. So we are not prepared to start with the drawing, the face folder for the linear system. I like to explain you yesterday just to remind you why we do that, because we know how to solve the linear system, okay. You heard from me yesterday, you can also talk in the morning about the exponential matrix, which is very forward to a method before any system of efficient, especially today with the computers, no problem. But why we do that, because it now wants to know how the face folder could store the linear system, because after that we don't apply that to the nonlinear design, it's might be the only way that we can come to some conclusions about this kind of nonlinear nonlinear systems. So of course we have to manage the Aden virus and Aden vectors, as I explained yesterday. Generally, first I suppose that the determinant of the matrix is different from zero, so that means that we have a unique equilibrium point. The other is very particular case is that I will talk more in the last chapter and I will talk about my location. But for now we only classify the unique equilibrium point, of course now it is the origin for the linear two-dimensional systems. So this is the classification, of course, it depends on the type of the Aden virus. Okay, we said that for Aden virus we obtain the quadratic equation, algebraic, so we have either two real roots different or we have a double root of real or we have a pair of complex roots. So let's start with the phase diagram type one, so we're learning the two different real and non-zero Aden virus. As I explained yesterday, we always start by plotting the phase curve for the canonical form of the system and after that we don't see how the phase portrait will look for the original system of the equation. So we have, so this is the canonical form, we have one the one and one the two on the diagonal, we have one couple system which we solved yesterday, so this is the general solutions of these systems of the differential equation and if for the original systems if we have Aden vectors we want to redo them, as I explained also yesterday the general solutions is given in this form. Here, of course, how the solution will behave depending on the lambda, okay, and the lambda is negative when the solution times zero, okay, and the lambda is positive when the solution is the exponential function, okay, it goes to infinity, okay. So we distinguish three cases and we have three different types of phase points. Of course, I start with the first case when the both Aden values are negative, so what we have, we have that the both coordinate of the solutions tends to zero. What this means, zero zero is the origin, this means that all the trajectories will approach the origin as the time increased, okay, so according to the stability which we defined yesterday, the origin will be the stable, but not only the stable it will be, remember asymptotically stable because all the trajectories are not just around the fixed point, it's approaching the fixed point as the time increased. The backward time, so as t goes to minus infinity, of course, the both solution goes to infinity. Additionally, both axes, so x1 and x2 axes are also the trajectories and the flow on these trajectories are also going to the origin, so we are approaching the origin, but now we don't have a curve in the trajectory and this is what we call the straightforward solution, so the line on which we approach the approach. So the next question is, for now we know that we will approach the origin and the time increase, but the question is how exactly will the trajectories approach the origin? So remember what we say, the trajectory is what? It is a curve in x1, x2, which is parametric expressions from zero variable solutions, so x1, x2, x3, we have the parametric equations of the trajectory, so in order to see how the trajectory exactly approach the origin as time increase, we compute the slope, okay, the tangent to the curving arbitrary point of the phase plane, so with quite easy calculations, so the first derivative of x2, across the first derivative of x1, we obtain the exponential function, so now it is important which of these two eigenvalues are bigger or smaller, okay, or which is closer to zero or, so for time, I suppose that a lambda1 is less, okay, less than lambda2, so where this goes, lambda2 minus lambda1 is bigger than zero, yeah, so this goes to infinity, okay, with increasing time or it goes to zero in decreasing time, so backward times, so what is the line with this tangent to x2, okay, x2 axis, okay, epsilon axis, and backwards time, the trajectories will approach the x1 axis, so this is the tangent, and we have the phase portrait lines, okay, so all the trajectories, so notice the directionals on the flow, on all the curves, we are approaching the zero, so anywhere when we start, okay, after some times we will approach the zero, the origin has the time increase, how the curves looks like, so we said, as the infinity x2 axis is the tangent, so the trajectories is tangent to x2 axis, what if we go backward, the trajectories become the parallel to x1 axis, so they are coming from the parallel directions on the x1 axis, and of course as I notice we have these four straight line solutions, this is axis, so when we start at some axis we stay at this, so we have the three line solutions and approach the origin also on this slide, so what type of this point is, so in the classification, the phase portrait is displaced, and the equilibrium point is called stable node, okay, so this is from the left side the stable node, or otherwise everything, I'll talk about that later, everything that is stable generally will become the sink, you know what is sink, and it's just eat all the triples sometimes, so the unstable six points will be called source, yeah the source because you look the picture on the right, this is just unstable node, all the triples just looks like the origin source for all the triples, so for the unstable node we now just go in the opposite direction, and unstable node is when the both eigenvalues are now positive, okay, so what is here important, we saw that it is important which eigenvalue is larger or bigger or close to the zero, and in this case we call the longer one the stronger eigenvalue, well you can guess, because the solutions will go faster or slower if the longer one is less than longer, so on that axis you will faster approach the zero than the other two, sometimes for some some systems define the difference, so how fast you will enough approach to the stable equilibrium, how much time do you take, so that's why usually we call in this case we call x2 as slow eigendirection, and x1 axis is called the fast eigendirection direction, so what we can conclude from the previous, so that trajectories will approach the origin always tangent to the slow direction, so always tangent to the slow direction, and in the backwards time it becomes parallel to the fast eigenverse, so the only thing that you have to, but to calculate in order to know how the trajectories look are so far the eigenvalues and eigenvectors, so this was the canonical form, what will change in for the original system, now for the original systems the eigenvectors are not the original vectors of the basis than the eigenvectors, okay that's why we did, so the stable note will be, and the unstable note will be like this, so you see we just changed the coordinate system as I explained yesterday, now the slow and fast eigendirections are what? Determined by the eigenvectors, okay by the eigenvectors, so the slow eigendirections is determined by the eigenvectors which corresponds to the what, so the eigenvalue which is concerted to zero, this is the slow part, and the concert part will be the one corresponding to the eigenvector which is far from zero, and again we have the same situation, here is the calculation in this case, but if you are interested in generally, as I explained it is enough to plot for the canonical consistent values, now you see we have just slightly deformation, so generally what we have to remember that trajectories always approach the origin tangent to the slow eigendirections, so defined as the directions determined by the slow eigenvector which is corresponding to the smaller lambda, an absolute value, and in backward time the trajectories become parallel to the always fast eigendirection, so here is the example, as I already explained, of course, we told you yesterday, of course, all the creation, even for the homework or everything else, now you can use anything that you use for mathematics, for example, for mathematics, math, math, everything, because it will communicate the values and then you have to put it out and so on, and the features are here floating more from the apple of the base vector, so on the left you see the base portrait on the canonical system, you see, and on the right is the base portrait of the original system, so what we have to do for the matrix, well to plot eigenvalues, now we have minus four minus one, and the eigenvectors is minus one minus one, so we have the fast eigendirection is the line epsilon equal minus six, okay, to equal minus epsilon one, and the slow eigendirection so corresponding to the smaller one, this is slow eigendirection, this is this line, so this is two lines plotted here, yes, so the green is the fast eigendirection and the red is slow eigendirection, so ICT, we have what, so all the trajectories approach, tangent to the red, and backward time return the parallel to the green line, which is fast, fast eigendirection, okay, and the third case so unstable is the same, basically now both solutions tends to finicky, backward time tends to too, so we obtain the phase border just by changing the arrows reversed, and according to the stable path, but the next case is maybe, okay, if two real eigenvalues are of the different sign, you know what, now the x coordinate goes to zero, but the x two goes to infinity, okay, so before I show you what is your guess, are we going to approach the origin or not? Depending on the direction, we go out depending on the direction, yes, but very likely depending on the direction, you have a direction that is stable, we are approaching everything, and the other is going the other way around, yeah, so that is what we have similar to the stable and unstable, no, we had there the stable straight line solution four, now we will have two stable and two unstable, but what about the rest of the trajectories, so if you start from the axis, you will either approach if you're on the stable axis, or you will move away from the origin if you're on this unstable axis, but what if you start from any point in the first part? We will not retail it, we will go firstly parallel and then curve, so yeah, yeah, it is a face portrait in this direction, so the origin is stable or unstable, you must follow enough time, just move away, away in the direction of the unstable axis, yeah, okay, so it looks like it's stable, for some time you will approach the zero, but sooner or later the unstable line will turn all the trackers away from the origin, so this is unstable, so the trackers look like this, okay, so in this case we have these two stable lines, okay, corresponding to the stable agent direction, which corresponds to the negative agent value, okay, and in this case the x2 axis is unstable lines corresponding to the positive agent direction, so if we start anywhere else, we will go sometimes parallel to the stable, it seems like that we are approaching the origin, but sooner or later we are repelled by the origin in the direction of the unstable line, this point, classification we call the stable face portrait, and as I just explained, this point is unstable fixed spot, you can maybe ask the question, what about the part about on the stable line then, that's why in some literature of the dynamical system this kind of instability is called stable instability, meaning that you have some position in the face plane, then if your starting lines are there, okay, then the face point will be stable, but if you start anywhere else in the face plane, one of the big points will be stable and we will move away from what will be different from the original, just again the transformation, okay, now we have also unstable agent direction, which is determined by the agent vector corresponding to the positive, and we have the stable agent direction which is determined by the agent vector corresponding to the positive, all the other directions will be the same, okay, again we are parallel sometimes to the stable and after that, no, I need glasses, which I am going in and out to the direction, so this is unstable, so we are parallel to the stable and then move away from the origin in the direction of the unstable, unstable direction, so this is the, of course you also have here, for example, of the matrix, the ideal value is minus 3 in 2, there is agent vectors and as I noted here, again with the green now, it is the stable line and with the red it is unstable line and the phase quadrant of the original system is blocked on the picture on the underline. So, okay, we are going to the second line, so when we have two real agent values, but equal, so now we either have this canonical form, which is the same as the previous one, but if we have the canonical form like this, then this is, if you remember the second system that I saw yesterday, so the second equation is a couple, a method of separation of variable and first one is then the winner, and it is very easy to obtain these general solutions of the system we have done this yesterday, so what we have now. Again, the phase quadrant will depend on, well, on the long run, okay, and we distinguish the cases where the agent values is negative for the agent values is positive, so if the agent values is negative, then again both solutions will depend to zero. Of course, it is pretty much easy mathematical calculation in here, and now to answer the question, how the director is again approaching the zero, they again thought calculate the angle, and now with some calculation we obtain it, so we only have t here, so this anyway goes to zero in forward line or in backward line, so what this means, so now we have only one agent-wide vector, okay, remember yesterday I told you, when you have a double rule, you have one agent-wide vector, to make this transformation matrix, we add the other vector by some other computation, but we only have unique agent vector, so now all the tractors we approach the zero, tangents to that agent vector, and we go backwards, the agent between the parallel to the same agent direction, because we have only one, this is the difference in comparison with the classical stable node, there we have two directions, we are approaching tangents to one, and moving away parallel to another, now we are approaching and moving away in the same direction, so that's why we called this fixed point degenerative node, in this case the stable degenerative node, and of course the unstable, so as we see, now we approach the origin tangent to x1 axis, but if we move away from the backward again from the parallel to that same directions, for the original systems we have more transformation, of course I give you all the necessary calculation in the text book, but I will not hear you spend the time on the calculation, of course if you're interested you'll come to that topic later, we can check, so this is the phase portrait of the original original systems, so now this blue line is in this case the stable agent direction determined by the only agent vector that we have here, okay, now and how the tractors look like, they here approach tangent, but if we are back hard time we again become the parallel to that the same, so unstable degenerative node is pretty much similar, so now the tractor removed away from the origin tangent to the unstable and in the back hard time way that is towards the origin also comes in the same unstable direction, also you here have some concrete example, I here plot unstable degenerative node, so we have here moon away, this green line is the agent determined by the agent direction here too and you see we have the tractors which are tangently move away here from the origin to that line, okay, in the the second case of the matrix, so for this case, okay, when we had two agent vectors, so the canonical form looks like this, so this is the general solutions, but now the tractors is nothing, so if you, okay, if you divide this, this and this goes to the constant, so we obtain but absolute moon is constant, so we have the straight lines, so all the tractors are on the straight lines approaching or moving away from the origin, so very very easy to calculate and the face portrait looks like this and we called the origin stable or unstable start node, also very easy to name for the for the gradient space portion, then finally third case corresponding to the complex agent, agent bias, the general solution is given in this form and you can guess what we all have now, now we have sinus and cos sinus, so we will now have the theory of the solutions of the system, okay, this is just the mathematical creation for these sitters, remember the canonical form looks like this, remember the real part of the ideal values of the main diagonal and the imaginary part on the right corner down with the change of the general solution is given this way, so this is arbitrary constant, we have exponential function of the real part, so exponential part of alpha multiple five c, so what we can say now military, so our distribution goes from infinity up to zero to dependent watts on alpha, okay, what is alpha, the real part, are the origin will be the stable or unstable, it will depend on what, on the sign of the real part of the painting, okay, only the real part determines the difference, if it is negative it will have the stable fix point, if it is positive it will have unstable, it will move away from the origin of the plane, what did you see here, now we are about to have a question, we will see if there is any other, yeah, I will give you an example, it seems that I mean that's not everything from the nature of the dynamics, you know very little screeners and the quality of space are stable, okay, so we don't know that, but any example where I mean that example and show that the parameter on is negative, so probably the dynamics because we cannot study the dynamics in the audience space and we have to transfer into the audience space, yeah, that's a nice question, the denominator of the dynamic is always good to calculate the original agent vectors, if you just want to precisely know how your solutions approach the only one, the stable models that you call there or unskilled, but in doing application, well, there will be truth under mathematicians, so not so much of the application, but maybe Joe can help me, it's very important in some application that we just do the phase portrait of the original system, not of the that in the canonical form, because we just want to precisely know how to try it, I mean in some applications, because in the canonical form we just have right, right, I am good, is it in some applications, I agree with you Joe, but be honest, I agree with you, because if you are here in this situation when the variables are concrete, in the original variable, in order to study the stability, it's easy not to do it in the, the study of the character, come to the conclusion about the students and the type of students, because in the canonical form, nothing will be known about it, if you are really interested in the part of the actual solutions, yeah, yeah, so we know in the next one, it's important, but in the other table, so if you are generally interested in the canonical form, don't go to fixed points on the solution, so our post-medial approach to the original text, can you be careful with the work, the re-ups, this is the video for script, that is why you do the quality theory of differential equation and not to get anywhere, one example of the stiffness is 180 values is very small and very fast go, and the average in the direction is close to the video, then the numerical methods can be really tricky, if you put the tracks after the study solution, I mean, tricky in the standard, you have to work with the other topic, you have to choose a very small staff, which then may come up problems, the accumulation of the error and so on and so on, that's why the quality material of the differential equation and this approach is better, so again, my question is, what is alpha means to do, we don't have this exponential part, we will circle around, that's what we will circle around, that's the kind of factor we saw yesterday on the dancing oscillator, the closed directory around the fixed point, that's what we discussed, but the standard will be what, I explained yesterday, it will be stable, but it will not be asymptotically stable, if you start close to the, you will just circle around, moving far away from that origin, but you will never approach exactly, so you'll have just a periodic solution, and generally, this is very easy calculating here, because the tractor is, so this is the parametric curve, if we calculate x1 square plus x2 square, we obtain the constant, so exactly with the non-occurrence, here is the circle, in the canonical form, but for the original system, we will have the transformation, because the axes are now different, that is the only transformation, so we have three types of phase portraits, the center corresponding to the real part of the eigenvector of eigenvalue is zero, we call unstable focus, so we have the spirals, which move away from the origin, so unstable focus, if the real part is greater than zero, and we have the stable focus, now the spiral tractors that are moving through approaching the origin as the time increase, of course for the original systems, the phase quantity generally looks like this, okay, just slightly transformation, the circles will be transformed to the opposite, and we have again the closed tractors around the origin, in the case of the center, or unstable focus, so I mean depending on the side, this is the example, so on the left you see the phase portrait of the center of the original system, so you have it, but I will just mention here, but I'll come back later to that point, if you look on my picture on the right, is it for now familiar, very similar to dating notes, yeah, you can almost no make a distinguish, if I just saw that picture, you will be completely right, I asked what type of the sixth point is here, if you answer this is the degenerative note, I will tell you, that's in your right, but not in the spiral, which is not in this picture quite well, but as you see when we now make complete classification of the solutions, it will become more clear, because in some sense, degenerative note is degenerative because, yeah, it has the property of the note, that does not correspond to the periodic solution, okay, it's a note, on the focus center corresponds to the periodic solution, okay, unstable focus on the center of the periodic solution, nor the applications, stable notes, stable dating notes are not nice, I don't think we approach to some fixed points, or move away, of course, I'm talking about disability, but, in a sense, the code is the case data and personal, yeah, very close, yes, very close, so the degenerative note is talking between the focus and the maintenance, we can also do that, so in that sense, so this is a concrete example, so the agent values are plus minus, if you have plus minus one, you have what? And you have alpha, alpha, beta minus beta, this is the picture on the right, and the next example, the agent values is three, yeah, minus two, yeah, so now the real part is negative, so we have the stable focus, so the spirits will approach the origin, okay, your face is, after all that, don't worry, the things are not so complicated as it looks, so we all have a very smiling face, the agent of the final session, so there is a really nice and easy way to make it complicated, to do that, the so-called trace determinant plane, remember, where is my character, here, the characteristic equation is here, I told you yesterday quite well that it will be at some point important how the equation looks like, so the coefficients are t, which is the trace, it's very easy to calculate, okay, and the other point is there for me, so you actually don't need to calculate agent values or the agent vectors, if you just want to classify your fixed point, okay, but if you want to more precisely describe the solutions, okay, then of course you'll have to do that, but just for briefly classification that we will very frequently done the no-linear dynamics, because there we will sometimes have three or four or five fixed points, so for every, you'll have to calculate, but this is the easy way, and you just really have to remember this smiling crazy face, there is everything that I was talking about, so what we do, we calculate the trace, okay, we calculate the determinant, for your matrix, you put the point in your td plane, and where you are, nothing but very very easy, so where is the difference, so what is this line, so this is the discriminant of the quadratic equation, yes, because of the sign of the discriminant, it depends what, are we going to have the complex or the width, or if the discriminant is zero, so if you are on the paragraph, then you'll have a doubling of the quadratic equation, so okay, so here, but this is the funny picture and this is the back of the picture, which explains it, so here, so above the, let's say, let's first say, below the trace axis, okay, so when the determinant is less than zero, the whole this line belongs to the saddle, so down you're on the saddle, remember what is the saddle, we have stable line and unstable line, we are going sometime parallel to the stable, but then we are moved away from the origin in the direction of the unstable line, so down we have the saddle, here, so below the parable, we have nodes, the trace is negative, this means that, okay, because from the weird rules, we have that the sum of two ideal values are the trace and the multiplication is the determinant, so this will give us a conclusion about the size of the ancient ancient values, so if you're below the parable and you're on the left, so if you're the trace or the matrix is negative, you have the stable node, on the right you have unstable node, on the parable, double loop, remember there is a degenerative node, also left file, stable, right unstable, if you're above the parable, you're coming to the complex solutions because now the discriminant is negative, okay, so above the parable, you have the periodic solutions of the linear system, on the axis you have the center, so here, this means what, this means that the trace is zero, okay, if again the trace is the negative, you have the stable problems and on the right you have unstable, unstable problems, or a spiral seam, or a spiral source, as I explained at the beginning, I'll come back to that point in the meantime, remember everything that is stable, we call it, the only thing, yeah, so the second quadrant belongs to same, and on the left, everything else is one table in this space, so everything else is closed, so below as I said is the stable and the first quadrant is, and now I'm coming back to the point of the degenerative node we see here, just as I mentioned, 5, 10 is below, so the degenerative node is where, on the border line, as you see between what, exactly between the spirals and the nodes, then it's white, intuitively very clear white on the face of this, how it looks like, but here I will mention one also very important point which will be very important for us in the nonlinear dynamics of the collater, this border line, it's very important, what you will say about this border line, border line between what, so on the left we have what, now it's white, unstable, so very important border line, you just need to be cross here, you are on the stables, but if you move away on the right, go to the unstable system, unstable point, so the center is, it has the stable, okay, as you can see from the border line here, between the stable midi plane and the unstable midi plane, this will be very quick in the nonlinear dynamics because the linear centers you will see can become unstable focused on linear, so in the linearization, you can move from the, if you conclude that something is centered, I will talk about that a little bit later, but the linearization will not give us any answer about the center, so if you conclude that there's something in the center of the linear systems, then in the nonlinear dynamics, then either the linear center still can become unstable focus, if it becomes unstable focus, then of course for the model, it won't have a nice behavior because we will not longer move to the fixed point, one of the solutions we are going to use all the way, and the model goes on stable in that direction, so very easy, if you just remember this picture in your head, then it's much easier to classify the fixed point of the linear system. So, okay, this is just in the conclusion as well, so if x square minus 4d is less than zero, we are above the power point, the real part is one half of the trace, so if the trace is negative, it is sinc, if the trace is positive, it is, so in general, if the trace is negative, okay, the matrix, you are stable or you have, you have the sinc, if the trace is positive, you are in the first quadrant and you have the other source, you can decide about the behavior, here is also one more picture, which I mostly remember because it is, it is most clear considering that the classification and the position in the right, and finally, according to the stability concepts, with the general model of stability has already done that through my classification, but let's conclude, so we said that the saddle point is stable, stable mode, stable focus, stable degenerative mode, stable star mode, or is asymptotically stable fixed point, sinc, and unstable mode, unstable focus, unstable degenerative mode, unstable star mode, because unstable fixed point or derivative sources. Center, once again, stable, asymptotically stable fixed point. Do I have one? We have, let's go right here, I also have assignments, I think it should be uproading, you can do exercises for, first for the structure, if I get you here the, no, yes, this is for planar system, okay, so find the, you know what, let's find the eigenvectors, and just going to that of this matrix, transform to the canonical form, and you can slowly sketch the face portrait of the original system and sketch the face portrait on the, in the canal, that's all. You can do it for homework, but let's do now just the most simple way, as shown in the last section, just make me the classification of the original fixed point. In order to, to plot the fixed point, you need a, again, okay, eigenvalues and eigenvectors, since it's all, but let's now just do the classification, just calculate the trace and determine and, and then we will see. Yeah. Okay, for the first matrix, so the trace is, trace is five, the determinant is six, where are we, a positive trace on the right of unstable, that's the determinant is positive, so we are up, so it becomes several, the determinant is negative, so for now, we can say what, what's that, but we have to calculate the, the discriminant, discriminant of the portrait, so the square minus 4D, 25 minus 74, it's one, the eigenvalues are real, we have two real, different, one side of the fixed point is unstable note. Yeah. For the second, the trace is, trace is four, the determinant is five, so minus four, now it's negative, now we have what, again, unstable, because the trace is positive, normally are on the right, but the roots are imagine, imagine, but now the determinant is complex, because it is negative, yeah, the complex, so we have unstable, unstable, yeah, unstable, spiral, we do what here, trace is now two, yeah, now the equivalents equal 4D, this is the parabola, so if it is equal, you are on the parabola, when you are on the borderline, still the trace is positive for you on the right, but what you do on the borderline is taking energy, but unstable, just keep the, okay, so now we are here, this is good, on the borderline, the trace is now negative, now the amount is minus nine, negative four, so it's going to be a tremendous level, okay, the D minus one, where we are, generally when I do the calculation, I first calculate the amount, because if it is negative, then I think it's a saddle, don't need to calculate, but if it is positive, then continue, continue, of course it's easy to calculate, so now we have the saddle once, okay, on the left it's nine, and the final is trace, minus three, this is three plus two, yes, so the quadrate minus 4D, minus this one, what will be now, so the discriminant is positive, which means what, the real, okay, the real line, so, but the trace is negative, so we are on the stable part, so now we have the stable, the stable part, so of course in order to opt more precisely the trace portrait, and of the systems, of the original systems, one of those systems, maybe you can do one example, at home just choose one, so that you can just get no interest or any calculation, but there are the classifications, that becomes, we very, very frequently use the classification between the countries that we will need to handle pretty much quickly the non-linear designings, because it is just one of the properties that we have to do in order to describe the base portrait and the system, so, okay, I think I will stop here for today, we are moving to the only non-linear line, let's see how it will be cast by the normalization, we're going to classify this one, but we will dance in some more techniques of research, the direction is more, maybe more important to do that, we have a limit cycle, in the non-linear techniques, okay, and the question starts now, okay, chat, thank you very much, so we meet in 20 minutes for the last lecture, of today, yeah, I don't know so but then starting from, I think, the work of mathematics, they input to the stream plot and you really do, you will see the non-linear dynamics, really do a nice job, even the limit cycles, which you have to do more, more, there is, of course, for my contribution to the non-linear, which is very nice for looking, but you can guess the various limit cycles, really do a nice job, so when you do just the stream plot, put the vector field, put the variance where you want to plot your approach, yeah,