 I'm coming from this university, I'm working at Faculty of Science and Mathematics as you see. I wish all of you welcome to our university, so I will talk these couple of days about the dynamical system, which is the main topic of my research work. I had questions when I started to prepare this course, but what's the best way to incorporate these dynamical systems in the main topic of this course? And the other question was where to start the previous knowledge of the participants of this course. So I decided to start with some basic definitions and introductions so that you can follow me in the last slide so don't worry. And since basically this is this one for you, not too much in mathematics, I'm trying to incorporate it in mathematics as I could, but of course, so what is the topic in the dynamical systems? I said that the differential equation is the field of mathematics which has a very long history, maybe over 300 years. And basic questions concerning differential equation and how to solve some differential equations. And the other basic question, at least concerning the mathematics, is the existence of the solution of course. So the question about the existence of the uniqueness of the initial differential equations, I'd say that this problem is pretty much so for a quite a huge class of differential equations as the first half of the 20th century. Thanks to the previously, the theory of the fixed points. Another question is how to solve the differential equation and the first we studied the linear differential equation. And the problem of solving linear differential equation was easy because the linear differential equation is constant or efficient using the matrix calculation, we can do that. And of the 19th century, let's say, the extensive study of the nonlinear differential equation study, basically motivated by the application. So in the application, more or less, you have a nonlinear differential equation. And of course it turns out that most of the nonlinear differential equations cause you to solve. And at that time, the point was in two directions. One is developed numerical matrix for approximation of the solution of course. And another topic, which is basic, which we will see in this course, I will try to present you the basic tools, is qualitative theory of differential equations. What we want to do, we want to characterize the properties of the solutions without having an analytical solution, and now we have the extensive solution of the differential equation. We want to describe solution as much as possible. So, for example, is the solution periodic or not. And we had also the resolution or not. Does the solution tend to infinity or tend to some of the things, limit is bounded or unbounded and so on. The field of qualitative theory of nonlinear differential equation, especially, was extensively funded in the 20th century and still is developing in that direction. So, I will hear this course to send you some basic tools of the qualitative theory of differential equation. So, of course, we will study with some basic definitions of what is the phase portrait, what we mentioned with the ability of differential to the dynamical systems. And of course, some mathematical parameters, since from the linear algebra, of course, what is the general solutions of the system of differential equation. Of course, here I will present you only the methods and techniques, nonlinear dynamics in the plane. So, that's how much I will have time. So these techniques can be applied for the high dimensions. I will tell you which one, but some of course is pretty much like it only with the plane, especially concerning the linear side. We don't start with the phase portrait for the linear systems with two dimensional linear systems. You will say, okay, I can solve it. Exactly know how the solutions of the linear two dimensional assistance space. So why to do the phase portrait, because we will use it in the nonlinear dynamics. So we have to know how the attractor is and how the phase portrait looks for the linear plan of the system in order to apply it in the nonlinear dynamics. And that's why the other way is okay, you will just see in the introduction that very easy to solve the system so differential equation. So we will classify the fixed points. And we will see basic three types of phase portraits concerning the two dimensional linear linear assistance. And we will see the easy classification of solutions and the next, we will continue with the nonlinear partner systems. We will see what is the linearization is part of course we will use the previous chapter so the case portrait on the linear systems. We will see how to construct the faceplate in the diagram in the in the faceplate with some with some examples and one of the characteristics of the nonlinear dynamics, which is not concerned with the nonlinear systems which goes away to the periodical solutions of the of the model of the systems that we care. So we will see just the most frequently used techniques, how to eliminate the limit cycles in some nonlinear systems or how to prove that there is a limit cycle in some nonlinear systems. And finally, I hope that we will have enough time. So what is the beautification, which is a very important application so what is basically the beautification, if we model anything, we will have some fun. If you change the parameters of the small changes in the parameters to the changes in the dynamics of the nonlinear systems. What is the change of the dynamics and the change of the stability of the fixed point. You can have one fixed point or there can appear two fixed point three fixed point or the fixed point can disappear from the changes of the parameters. And then you all change the dynamics of nonlinear systems that's what we call beautification and I will present you the three most difficult several transcritical and before verification which concerns to the appearance or the appearance of the fixed points and the hope of the verification which comes down to the appearance of the nonlinear cycle in the nonlinear systems. So that is our last topics for the course. As I said, I decided just to start the introductionary as we will see maybe we can go faster or slower to this part. So, of course, what is the systems of and differential equation. So for this form, the function f is the given function and unknown functions are we have unknown functions, epsilon one to epsilon and and T is of course, the parameter of the function. We use the simple notation, we use the vectors and the matrix. So, so complicated so we can read rather than the system in this this form using the vector vector notation of the vector and of course the vector field of the capital F. We will hear concern. Something that is calling differential equation out on systems of differential equation. What's that mean that mean that here, the right side of this function does not depend on the time. You will see the advantage of this, this particular for differential equation. Of course, we will start as I said with the linear differential equation here you see the system of linear differential equation with the constant coefficients, constant coefficients, but we will put them in the second order. Really, first of the differential equation, you can solve the linear differential equation, you will see like, basically, in the differential equation, first of all, we have no problem. Even the second order in a differential equation. And this is all for all the coefficient only the linear differential equation with constant coefficient, we can solve using the matrix. So, okay, the solution. Here, the initial problem so what is the initial problem. So, if you win application now, the, the solution at some point of time. Of course, we find zero. And using that initial condition, you want to solve the differential equation or to see how your model will behave or as time passed to starting from the initial initial point. And the kid from mathematical points of view but you will see here also we must have a conditions for different than unique solution, you will see like one very important reason. In order to plot the phase diagram. There is, mathematically, many different types of theorem of existence and the difference of the solutions on the different conditions, but the most recommended use the question application. So, what do we need to have existence of the solutions that the functions on the right side is continuous. If they are, you have consistent solutions but not yet. So, what's that mean that mean, two solutions to one point. This is very difficult if you go into the numerical summation, because you don't know which solution you have to try to solution pass. And, of course, it is not, not good also in the quality theory of differential equation. So, basically, if you start with the numerical methods or the quality theory of differential equation, first that you do is to assure that you have existed so on the solution. Basically, this is a mathematical problem or less because your model, almost anything that I'm not interpretation. It's very, very rare that you have function, which is, which does not satisfy this condition. Actually, we study, as I said, two dimensional, nonlinear system of this form. Let's add, we cannot solve it. So, what we want to do, we want to describe the solutions, the properties of the solutions, only by functions F and G. And you will see how we will actually do that. Of course, again, we use this notation, X is a vector and F is what we call the vector field. Of course, the fixed point of the system or equilibrium point or stationary point is a constant vector which solves the right hand side of the differential equation. We will start with the planar differential system of differential equation with a constant coefficient. So, in the next chapter here, and again, using the matrix, which coefficients are A, B, C and D. Again, we can write the system in the matrix form, as we said. So, very simple, X prime is a matrix of the vector. Of course, if the determinant of the matrix A is different from zero, then this system has a one equilibrium. Okay, because for the equilibrium, you have to solve this linear system of differential equation, homogenous linear system of differential equation. The right hand side of some non-linear systems, so here the function F and G, defines what we call the vector field. Great. Moving too much, yeah, the vector field. So, at each point, the mapping F, which is defined by small f and small g, assign the vector. So, think about what is f of x. So, we can consider it as the vector, which x component is f of x and g of x of epsilon. We visual the phase, based at the point of x epsilon. You can now see that for the system, very simple, which is mounted here, the vector point is this point here. So at every point, we draw the vector. Okay, we know the vector, is it clear, because the vector is given only by the differential equation. By the function f and g. So, in every point, one, two, we just calculate these functions and we have a vector. So, to every point, we can point the vector. So, we can point the vector field, very easy, and almost all the computers program here use the law from a mathematical basis, and also, you can do that pretty much easy to the vector field. So, what is now the phase quarter, so just imagine that you flow to the vector field, so just the approximation of the vector field. Just imagine that you start to know the vector field like this. So, if you're flowing along the flow along the vector field, the phase points, what to do, it's trace the solution of the system. And we have the trajectory, what we call the trajectory. So what is the trajectory. The trajectory is the curve now in x epsilon, which we call the base plane. And it is the curve which parametric equation is given in this way. So, actually, for all T, so for all time, the target vector at some point, we know that's given by this or given by this, it is based in this. So, every solution of the differential equation in this sense defines a motion along with the curve. So, drawing the trajectory in x epsilon plane, which is, as I said, we will call it the phase plane, and but we don't only plot the trajectory, we also track the directions of the trajectory as the time increase. So, we plot the trajectory together with the motion along the screen with the increase in star, and that is what we call the phase of the system. So, this is the set of all trajectories in the phase plane. Geometrically, the dynamical systems is describing the motion of some points in the phase plane along the trajectories, which is, as we saw defined by the systems of differential equation. So, you probably asked how we will do that. That's the main point that I will try to show you how to plot in the phase plane, because for the linear systems, we know the solution. If it will be easy to plot in the phase function, we know how the curves looks like, but for the linear systems, we will see that we can use the techniques. And what can we say about the solutions after that, that we work with them. So, by describing the phase plane point, you can say the solutions are pretty. If the solutions tends to infinity points, that means that we will have some fixed points with all solutions to approach as the time increase. So, that means that the solutions will go, or we will have unstable fixed points, that means it will mean that your solution tends to infinity, plus infinity means that it will be unbounded. And you will not have stable dynamic assessments. So, for the non-linear systems, the phase portrait would look like this, for example. So, we will have the fixed point like this, like A, C, or B. If you now look at these fixed points, we can say that all these three fixed points are unstable, while except this, this is very particular case that I will talk about a little bit later. So, only if you start from here, you will approach to these fixed points. If you start anywhere else, you're moving away. So, from everything else, you're moving away from this, also from this point, also from this point. But the fixed point B will have some interesting point to application, because here we will have a periodic solution. We have the spirals that circle around this point, and what else we will have in a non-linear dynamic, but not in a linear case. So, this is close curves, like this D, which, as you may assume, is characteristic of the periodic solution. So, if you start at this closed circle, just circling around with some time, you are coming back. So, you'll have a periodic solution of the model that you will have. So, the cycle, the issue is the periodic orbit or any closed trajectory, which is, of course, not an equilibrium point. So, I already mentioned stability just on this picture. Of course, generally, the stability is if all the trajectories are approaching with increasing time to some point. So, generally, the mathematical, formal mathematical definitions here. So, what will be the stability, but maybe better explained on the picture, of course, you have the formal mathematical definition. So, if you start close enough to the fixed point, you will stay close enough to the fixed point. Okay, so you will not move away. If you start close enough, that's me, from here, you will stay in some circle. Big girls, small runnies, never mind, but you will stay close. But except of the stability, we have also the asymptotic stability, which means what? Not that you will not move away from the fixed point or from the cycle, but you will approach. So, not really that if you are starting close to some fixed point, then after some time, you will approach to this point, which generally is not to this. So we have the fixed limit also expected to that fixed point. So, okay, maybe one basic, very basic example of the second order linear differential equation, which of course can be written as the systems of the differential equation. Okay. Are you familiar with? Do you know the differential equation? Okay, it's already here, here. So this is the replacement. We have the mass attached to the spring and x denoted the placement of the mass from its resting place. So, x is positive, is stretched, and it is in this contrast. m is of course the mass of the oscillator, damping constant, and the spring constant, and if b is equal to 0, then we have this very simple differential equation. This is second order, it is linear, this is constant coefficient, and we said it is a homogen, what is a homogen that means that the right hand side is 0, you can also have the function under the right hand side. So, written in the system, this is of this form. So, let's see first how the vector field looks like. We are on the, we say that we will trace the vector field, the vector field we know from the right hand side. So, let's see for this example how the vector field looks like at each point. First, let's see for 0, that's mean on x axis, okay. What is the vector? The vector, the first coordinate is 0, and we have the second one. So since the first coordinate is 0, the vector is like this, okay, upward downward. This upward, if this is the second coordinate, so this is positive, okay, then we are going upward. This is negative, we are going downward. So, we are going upward here, or downward here. So, how the axis look on the V axis, so the vector field is now V0, so it's horizontal, okay. So again, it is positive, the first coordinate is positive, it's going to the right, and the first coordinate is negative, it's going to the right. So, here we have to the right and down, so below the x axis, we are pointing to the right. Of course, I will show this will be one of the basic techniques how to plot the vector field for the nonlinear systems, and we will of course do this more carefully, because it will be related probably, probably tomorrow. So, but at least from these vectors, we can now imagine how we will circle around. The 0, 0 is a fixed point, okay, on the system. And if you start there, you will stay there, the fixed point is like that. You start your motion from the fixed point, then you're moving, so you're staying there, but if you start anywhere else in the face plane, you're just circling around the fixed point. You have closed orbits, and this is definitely the face holder for these systems. How can we interpret this in the motion of the mass on the spring? For the fixed point, this is equilibrium point, so we are not moving, it is in the rest position. What about the circling at the close orbit? So, when the placement is negative, so we are here in this position, or in the face plane, we are here, okay. So the placement is negative. Yeah, there is a question. Yeah. Fixed point. Disorbit. Stable. And what you mean? Because all the three orbits. There is an extra region. No, no, no. Fixed point here is stable. Because if you start closing up to these six points, you're staying there. You're not moving away from the fixed point, okay? But the closed orbits don't define the stability property of these closed orbits. You are just having the periodic solutions. You can also assign the stability property for such closed orbit. But do you have something like this? Yes. Yes, because the initial value is nothing else than the fixed point. So, you're starting from one fixed point, that means you're starting from one circle. But there are no speedo's basically. For this, no. Yeah, no. No. Once you start with this, you are using a very near orbit, near the orbit. That is actually coming to the orbit. But for here, I'm using the initial area, that means complete the full circle. So, I have 7 unique orbit. Yes. Then how can I state this orbit in this fixed point? Because I have 7 orbit. Yeah. The initial condition is that if you start closing up to the fixed point, then you're staying closing up. You're not moving away. If you're just circling, you're not moving away. Just imagine that this is the regular circle on the left side on the pitch. If you are on the circle, then you are always on the same distance from the fixed point. Okay? You're not moving away from the fixed point. So the fixed point is still stable, but it's not on top of the fixed point. You're not approaching. I will talk about this also later when I classify the fixed point. This is the fixed point, which is called the center. This is the only fixed point that is stable, but not asymptotically stable. You will never approach this point. But the movement is the same. You're not moving away from this circle, but you're not approaching it. It's stable, but it's not asymptotically stable. Anyway, that's a good question, of course, because this is a very typical, very typical analysis. You will see that the center will have very typical in the application. But also, it will make the problem in the study of the moment. It will not be very easy to establish that we have such a typical failure of the moment. I think that's a good question. Sorry? Sorry. If I'm not wrong, I have a new field to be asked. It doesn't show the characteristics of the system. Baseball? Yes. What is your, what do you think it doesn't show the characteristics? I don't know, it's called zero number. I don't know. Anyway, she just got the probability. Yeah. So, okay. So the first position and then the placement is negative, but the velocity is zero. So we are off and the screen is most compressed. Next, phase point of this direction. So we are moving in the clockwise here direction. This way. So we are moving along the close over and X increasing, but V is now positive. So the bus is being pushed. So we are on the V axis and the position B. But at that time, the mask is rich. So X is zero, but at this point it has a large velocity. So over shoots. And we have after some times and the position C. So we're now the X is positive. So we are now here. The mask spring is batched down. And finally, must get pulled up and eventually completes the circle. Start position to D and from D to A. Again, up going up. Okay. As I said, I put just the basic tools that I'm going to use, but you probably can I go this faster. So what actually we use even more than the vectors. Okay. No, the non zero vector, which is called a vector. So we have a selector satisfying this equation. So, of course, this definition is also applied for a high dimension one here. As I said, we are doing the plan. For the matrix. So how to find the sets also are following the characteristic equation. And in our case, the matrix is ABCD. So the characteristic equation is pretty much typical we will use this later from that's why I wrote it. So this is basically the quadratic equation. This coefficient here is the trace of the matrix. And this coefficient here is the beginning of the map. The calculus is pretty much deeper. Of course, the again, the egging bias depends on the square minus 4g. So this command, of course, we will, and the later consider three typical face portraits, depending on the nature of the roots of the characteristic equation on the egging values. So we have two reels and this thing, or the simple real root or to conjugate complex root and of course the appropriate egging values. Of course, solving the linear systems of differential equation of second order of high order. Never mind is done if you find the egging values of the matrix. So the linear systems with the coefficients of course. So how we do that so if V is agent vector of semi matrix associated with the egging value of lambda, then this function, so exponential function. A lambda t multiplied by the agent vector is the solution on the differential equation. So, like here I show this but this is really pretty much basic. calculation and very easy. Of course, using here that V is agent vector so here and up to that it's pretty much. So, for our matrix two times two, we can find two egging values so we have two solutions. Okay, and if you find the egging values which are different than the appropriate vectors taking vectors are what we say the linear independent and for the basis in the in the face plane or in the in the plane. So we are in our collaborative which means that I will go to this of course. I will send you after I finish so from today you will have this text book, which is here. So, if we make what we call the linear combination of this two solution. We can also find that some arbitrary real constant. So alpha multiplied by one solution by the second solution. That means what we call the differential equation the general solution what the general solution is means that from the general solution you cannot pay. The initial value problem, you just change the initial values and that means that you will calculate the constant alpha and beta. So they want arbitrary but particular constant and then you obtain this solution also initial value problem. So, I'm obtaining a very nice negative values are very close to solutions or from the differential equation for the general matrix. There is a lot dimension, we can do this way but there is a easier way using exponential matrix, which is equivalent to exponential function. This is the first order differential equation, but this is not going here only what we'll need. Of course, if we have some typical you will see later matrix form of the second order differential equation. So, you can solve it easier. So, no need to make him to obtain the day you have to send a device of the system. And one of such system is what we called you. Okay. system. Basically, the more simple that in the first equation, you can just separate these two equation and store it separately like the first order differential equation. So, this boat to this first order differential equation is what we call the by the method of separation of variables. So much easy as you see. So, X on one side, T on other side. Okay. So, this is the X. So you move X on the left, detail on the right, put the integrate. Pretty much. This is the first differential equation. All the basic courses. The second one is the differential equation. We have the general solution, of course, also here the C is arbitrary constant. But if you have some initial value, X of zero, then it will be equal to C is basically determined by the initial initial value. If you have some initial value, then you will have the constant to see and you have the solution on the initial problem. So, using this, you can now be much easy without any problem. So this is what we call uncomfortable inner system of our second order. So this is the solution of the first equation is the solution of the second equation. method of separation. The second example is like this. Now you have a lambda on the main diagonal and one zero. So the system looks like this. Now it is not uncoupled in the second equation is solved by the separation. Okay. But if you know epsilon, then the first equation becomes a linear differential equation. So the general form of the linear, first of the linear differential equation is like this. X prime plus some function multiplied by unknown function is equal to some given function. We want to solve the text. And we have a formula, as I said, so this is the only linear differential equation. In the second order, linear differential equation, arbitrary coefficient in general, it is not solved. So we can only do some quantitative analysis on numerical methods. So I gave you here the formula. If we apply this formula now, of course, P of t here is only the constant lambda. The integrals are much easy. Okay. Here Q of t is epsilon. Okay. So we first solve this, we have epsilon, and then Q of t is just epsilon of t which we obtain. So with these integrals we form the general solution. As you see, the first equation is just the linear function multiplied by exponential function, and the other solution is with exponential or the exponential decay, depending on what that's what will be important, important for us. So why I give this particular example here to show you how to solve because we will use it to plot our first phase function, namely, for any given matrix, two times two. There is invertible, what does mean invertible, there is inverse matrix. And that we can transform the matrix to one of these three forms, three forms, actually. That's not many differences in two or three, but I will explain that later. So every matrix you can, you can transform either to this form. You can do a real constant on the main diagonal. If you do that, then this is our first example. So this is uncoupled system of differential equation for variance. The second case. And it's also our second example is lambda lambda and one. So this means that the second equation is solved with separation of variables and the first equation is linear so we can. Yeah, so I didn't want to go into the solution of this force type. The lack of time and the technique is pretty much different, but you can also be much easy, so also the systems if the matrix looks like this. And the matrix will look like this if you have. Yes, what kind of engine once. The first case and you have two different. The second case is when you have double root, the lambda lambda double root, so the matrix with the look like case two or three, you will see how it depends. And in the third case, you will have a complex. This is exactly the only three possible ways of the. And for each, you will see that we will pretty much easy obtained the face culture. The problem with the third case is not the complexity of the values. The complexity is the problem of the magic is not diagonalization. I mean, in this way. But you will always get on like this. So the real part of the main diagonal and imaginary part of the wilders here, but with the different side. I was talking about the third case number three. Number three. Yeah, you have problems because the matrix, you can diagonalize it. But you can always put it in this one. So, yes, that is the next technique was how to put the matrix in either of these four forms. Okay, as I said, generally, it depends on the agent virus. So just the show. So what we do for somebody that don't know what is the, you know, what is the genre. Okay, what we want to do, we want to change the coordinates. That's the most simple explanation that I can give you. We want to change your the coordinate systems. So to put the metrics in this way how we do that we make some what you call the mathematical linear transformation what is linear transformation that is not. So we want to multiply with the metrics that's the metrics. So the number has to be in vertical so it's different from zero. And if you multiply from the left side with inverse matrix and from the right side with that matrix, you will put the matrix a, in either of these four forms so let's go. Okay, before, before the example. So what this actually has to do with our differential equation. So we also transform the differential equation, but we'll change nothing that you see almost not really just obtain the secret form of the matrix to plot. That's, that's our, that's our problem. But as you see, we will not change, we will not change the stability of the fixed point, which is the most important thing that we're doing with the stability of the fixed point, we will not change the properties of the track. If you have a fixed point, it will stay the same. Let's say that the phase program that you will see, we're just slightly default. The formation is quite logical. If you think that you don't have no more. The systems like this. Now you have different coordinate systems, which is not a particular that is basically what we do we change the basic way to solve the system. So, if we transform our leader systems. Now, instead of metrics, we have this, what we call, I called it here canonical form on the matrix. What is what the colleagues said, especially in the mathematics, this is show the canonical form of the matrix. And we can also do that with more calculation and more mathematics for any, any matrix dimensional. Dimensions. Yeah. No, it's just diagonal. So, you have on diagonal some values or the blocks of custom metrics on the diagonal. They're not connected with. So, what is the connection between the these two systems. As I said, if we have the systems, let's say, of this new system, then X of T, so multiple by matrix transformation P multiplied by solution of the second gives the solution of the first. Okay, so if we saw first, and if we have the matrix transformation, then we have the solution of the original system. Okay, and as we see, the second system is very easy to solve in first three cases. And it is easy to solve the also in the fourth case, but we will, as I said, not go there. So it's very, of course, in the contrary, also if you have the solution from the matrix, the linear math theory or the special also conveys the solution of the original systems to the solution of this system, we will call it in the canal. So it will be probably easier if we go to some easy example so what is question for the first question is how the matrix. And if you have matrix T, then we just obtained the reverse matrix or multiply this matrix T minus one and obtain this canonical form. So the question is how to what what's the form of the matrix T, the matrix of transformation how the linear transformation looks like. In the first case, when the 18 values are real, we have two eigenvalues. The matrix T is very easy, just put on the columns the matrix. Okay, so if we have a eigenvalue vector here and the other eigenvector, the matrix is, this is the first vector and this is the second vector. Since the vectors are independent, the matrix is invertible. T minus one multiple A multiple T will give the canonical form of the system and canonical form will be really just obtain the form with the real eigenvalues of the main vector. It's always like that. Basically, you can write this matrix without T. Okay, because we know that you can learn that you can always write matrix. But what do we use matrix T? But we need the matrix because when we go to the solution of the second, we have to go back. So if you have the solution of the systems in economical form, you want the solutions of the regional system, that's why you need the matrix T. Otherwise, you will, you can write the form of economical matrix without matrix T. It's always like this, no problem. In the complex, then what's the problem, not the big problem is that when you have a complex, thank you Alice, you will also have a complex thinking well. How do you form the matrix of transformation now? You put the real part of the eigenvector and the imaginary part of the life as the vector of columns. Okay, so the matrix T will be matrix. And the canonical form. This case looks like so always a real part of the eigenvalue on the main diagonal and imaginary part on the other diagonal but with minus and plus. Maybe the most difficult case is the repeated eigenvalues. Why? Because if you have a double root of lambda only, then there is two cases. One, sometimes you will obtain two eigenvectors, which are ordinary values, but mostly you will only have one eigenvector. So how to make another, so to make the matrix of transformation. So the matrix of transformation will be this eigenvector will be the first column, but how to obtain the second. So the second linear independent vector will be obtained as the solution of this system. A minus lambda, lambda is the eigenvector of V, or I just wrote it in the usual matrix form. So you just, you have V, V is the first eigenvector, so you just solve this linear system of the equation according to double one and double two, and you obtain this second vector and make the matrix transformation, which will transform the matrix in the canonical form, which now have lambda and lambda on the main angle and one above the main diagonal on the right. So here, here I have a couple of examples. The first example, you have a matrix. So to find eigenvalues, actually you have determinant of this. If you have the quadratic equation like this, which solutions which roots are minus two and one. This is the first case. When you have two real and distinct eigenvalues, to find the eigenvectors, what we said, so you solve the system. So for lambda minus two, we have this equation. Of course, now it is actually one equation because the rank of the matrix is one here. Is there any cases for you that we must not exist? Is there any cases for that we must or not? Here it always exists because the format of the linear independent factor, because the thing that always is the determinant of the absolute. That's why it is important that the matrix C is form of two linear independent vector. This is so then the better. So that means always do the bit that in terms of cases we don't do that. That's why it is important that the vectors are. No, I mean, this is generally in connection with the matrix. The matrix. Okay, and we transform the matrix. If you have no linear systems, what? Yeah, that's actually what we will do later because when you have a nonlinear system differential equation, when you do linearization, then from the nonlinear system of technical linearization, we will make a linear system then it will be possible. And you cannot paint the structure of the base for around the very close around the global picture, but we will talk about this in the continuation. So, okay. Right, so in this or this. In the first case, one one. The second case, for example, to one. So the matrix transformation is this is first vector. This is the second. So the order of the vector so not before you can put the first in the second, then in the diagonal form you will get. The first day of the life on the right corner, but it is not much. You don't have to pay an action about the queue of the queue. It's not the same. It's not the same because the Jordan normal form, not necessarily only make the one is a diamond can have blocks, which can look like here. So the wires and above the main diagonal, you have one, one, one, one, one. Depending on the, depending on the is this room. The dimension of the state on the number of the 18 vectors. For the when you have the of the then the subspace of the in the last only the number of eigenvalues different. No. Yeah, the number of the multiplicity of the rules of the values coincide with the dimension of the associated space of the. And if not, you want the best thing you can do to create a yellow. Yeah. But not necessarily just the form of the conversation will just rise on the main guy. So with this transformation matrix. As we know, we have this form so this is one of the one on London to make that. If you just change the order of this in the middle of the one here and mine here. Okay, the next example is the complex. So in this case, you have roots. Plus minus one. And to find the a directors. We again. Change here. The agent value. And obtain this system which are reduced to one equation, which is there. So we don't want one so we can take her for example, I took that we want is mine. And then we do is minus square or one. Okay. Now, as I said, you obtain a complex. Okay. Now you have to a part real and imaginary part. What is the real part of this vector. Here is zero. And for the second real part is one. Okay. What is the ordinary part here was minus one. And the second part is zero. Okay. So just separate real and imaginary parts of this vector, because now we will make the matrix transformation. This is the first vector. This is the matrix in this case. You will always get the complex, but you separate the real and the imaginary part, and the real part will be the first column and the imaginary part of the agent vector will be the second, the second call. And this search case. Now we have a double. This matrix. This quadratic equation has a double root minus four. If we try to find the agent vectors. We can obtain one equation. So we have one agent. So I put so we want minus one and then we do this. So we need another. Okay. Another rating vector will be the solution of this system. So long days minus four. So you just end in more or less you use this matrix. This is divided by double one double two. And from the right side is the real agent vector. Okay. Minus one and minus four. So this is again just the simple inner system of job record to equation again the rank of the matrix once we obtain one equation. Of course, and here is the question the solution. So you just choose double one and you get the second vector. So I hear choose double one, one quarter. And then so now the matrix transformation is this is a real agent vector. Okay, one that we obtain. And the second column is the vector that we obtain this way. With this matrix transformation. Now the canonical form looks like. Which minus four minus four. We have one. So this is that second case that we also saw. We will see that generally this, this technique is used as I explained. So we will have a matrix. We will transform it to canonical form in order to easily. And first of all, so you will see that the base portrait of the canonical form base portrait of the given system so it will be We said the logical equivalent of not seeing the expression around a change. It will be always enough to put canonical form, which is very easy to conclude whatever you want. So tomorrow, sketching our first phase portrait for today, I think I will finish the questions. What is the minimum requirement for a full day period. Other questions. Thank you. I mean, systems. That's why we do everything. Yeah. I mean, for the equation for the later question it's easy. You can, you will see it. It will be very easy when you just respond to the matrix. The end in two minutes. The base portrait of the canonical form. The biggest problem you will see at the end in the nonlinear times is exactly how to prove that the nonlinear system had a minimum cycle. It has a very odd distribution for certain systems very, very hard. Even to the magic, not, not to mention how you'd imagine all the systems to prove the existence of the limit cycle, meaning you have a periodic solution. It's not very complicated. You have to prove the variance, the variance, the variance. Of course, you have to make an attractive. Yeah. Sometimes it's your 12. That's why that's why we learned how to defecate because traffic region. Yeah, it's tricky. It's tricky. Yeah, but of course this is the basic tool that I will explain you. I think online. Thank you. Not so much, but I try to do as I said, as the last method that it says, I just want to go deep inside the mathematics to what is the basic mathematics calculus that we need you to continue. Thank you. Thank you. We'll first stop sharing and let's thank you. Yeah.