 Yes, now this stream is yours, as you mentioned. Yes. Okay, thank you very much. I think that you should take an advantage of keeping a microphone and camera off. You should drink, eat, enjoy. I know. And sometimes if something is unclear, please place questions into the chat and immediately sip your coffee and eat something that you have with you. And we have to do it under some. Okay. At least the sunshine, correct? Correct. And now I see the nice question about the butterfly. Thank you very much for this. And let me stress your attention that the shape of the butterfly is because of the sun and not because of the earth. The length of the rotation period is this 27 days is because we observe it from the sun. But the butterfly diagram, the shape of the diagram is because of the processes inside the sun. There is something inside the sun that is responsible for the appearance of the sun sports at different latitudes. So sometimes there are indeed located only close to the equator. Sometimes they are at the mid-latitudes. So it's only the processes in the sun, inside the sun. I lost my presentation. So I posed the question, will anomalous growth in the instantaneous period results in the desynchronization of toroidal and poloidal components of the solar magnetic field nowadays. So let me return with her moment. Let me return to the previous picture and to recall that in addition to two anomalies, which we observed with a naked eye with solar proxies, we see probably another animal. We don't know right now just looking at this graph and what is more important that the solar proxies themselves do not expose an anomaly. It's the right anomaly nowadays. So we will try to have another look, a look from another point of view. The mainstream is, you probably know, is related to partial differential equations so called MHD magneto hydrodynamic equations. They are very, very complicated. First of all, even before you want to solve something, you have to formulate the problem. And even this step is not very clear because you know the equations, you know them from the first principles. But equations have coefficients. How to get the values of the coefficients. You can measure something and derive the coefficients from your measurements. But you can agree that it is not easy to measure something inside the sum, but this is the part of the probe. Okay. Somehow, scientists are able to write the equations. You have to solve them. It's difficult to solve partial differential equations, even numerically. First astrophysicists perform huge simplifications related first of all to some symmetries. You can assume that the sun is a bowl to some extent. And simplify the equations. On the other side, we do know that observations are not symmetric. We know different types of asymmetry. Axial non-symmetry. They are implemented into the system after this simplification of the equations. So you can introduce this non-axial, sorry, introduce axial non-symmetry into coefficients of the simplified equations. Then you have opportunity to solve them numerically. And this is the mainstream. We can discuss how good or bad this mainstream, but this is, you know, this is what we have to do. Because this approach comes from the first prince. Alternatively, you can mimic your observations with simple, so-called ad hoc models. And if you can reproduce these such simple models, observations, the dynamics of solar activity, then you can get some new information. You can make some judgments about the sun, about solar activity. And this is exactly what I'm going to show you with so-called Kuramoto model of kaotong oscillators. Well, I am very sorry for this slide, for the equations, because as a listener, I would prefer pictures or movies regarding the sun. But this part is technical, but honestly, I will show only these two equations. Many times, but these two. So by the end of the lecture, you will be quite accustomed to these equations. Let's have a look at them. These two equations show us that the derivative of the phase theta prime is equal to the natural frequency of an oscillator. Omega plus the term, the second term, that couples the two equations. The coefficient of the coupling is denoted by K, K1 of K2. The coupling, this sign is nonlinear. You can say we expect that the phase difference is small. So we will turn to a linear equation. But it is not the case. You remember the phase difference between solar process. It is not a couple of days. The scale of the phase difference is closer to a year. So it's not small number. It is some number. We will see that the phase difference stabilizes under some natural assumptions, but stabilizes to non-small value. And we cannot, cannot linearize our equations. So to summarize, we have two oscillators, each equation represents an oscillator, and they are coupled nonlinearly. We are going to consider the model with different K1 and K2. So oscillators influence each other non-symmetrically. Now notation. We will use the phase difference called here theta. And we introduce the average natural frequency, which is half a sum of the natural frequencies and delta omega, which is half difference. Similarly, we introduce symmetrical and asymmetrical components. It should be, it must be components of the coupling K and delta K as half sum and half difference of the coupling coefficients. We will see that this notation has a clear physical meaning. Now we start with the Kuramoto model, I just explained. And after the change of the notation, we can sum up and subtract the two equations, and we get their equivalent modification, equivalent formulation of the problem. Theoretical fact. The coupling is strong. So the equations are coupled. You know, they're forced. Then the coupling is strong means that symmetrical component is large with respect to two delta omegas. In this case, the second equation has a stationary point, and this stationary point is stable. So as time goes on, the phase difference, as I told you earlier, stabilizes to some value. So in our equation, close to this steady state, the left hand side of the second equation is approximately zero, whereas the left hand side of the first equation is approximately a constant. The direct problem is to describe the stationary state, given all the coefficients. So we know K1, K2, so all k's, all natural frequencies, and what is the limit phase difference. But it is nothing to do. Nothing to do because it's, you know, it turns to algebraic equations and it's very, very simple. I'm sorry, for a moment I stopped sharing, I lost a part of my cream. One moment. It's my technical thing. Now everything is good with me. You share in the presentation mode, yes, thank you. I need another mode to look at the other part of the screen. Okay, so this problem is trivial, it's nothing to discuss. But we are interested in the inverse problem. So we know, we know the oscillators, we know the solution. It means that we know the left hand side of the yellow equations. We assume that the oscillators are synchronized to some frequency and we know this frequency center. We know the left hand side of the first equation, which is the mean instantaneous frequency. And we know it because we know the data, we know the solution. We also know the natural frequencies, omega and two delta omegas. And the question is what are the coupling coefficients, k and delta k. Or we know k and delta k, we know the coupling, and we're interested in the reconstruction of natural frequencies. We will focus on the first problem. And it's also in theory, it's nothing to do, because we have two equations and we have only two unknowns probably I have it's shown as the next screen. So we know the left hand side, as I said, because the left hand side of the second equation is approximately zero, we can observe the left hand side of the first equation. And if we also know natural frequencies, then we have only two unknowns and two algebraic equations. So it's nothing to do in theory. However, let's relate, let's return to the sound. What is the problem? Let's do it step by step. First of all, what is the input of our model? We can use, as I said, two indices, indices that I have shown you, smoothed ISSN and AA index. This input is not marvelous because of the noise, but we can smooth it first, say, within four years sliding window, and use these two curves as the input. The red curve is quite regular, the blue curve is not so regular, is less regular, and what is important, they do demonstrate a phase shift. So we can assume, as in theoretical model, that the two curves exhibit synchronization. We will return to this point a bit later. But what I'm going to show right now, that the correlation between the two waves determines the phase difference. Remember that in theoretical model, I said that if we have two sine waves, we know the phase difference between the two selectors. How to compute it? The practical way is to find the correlation between the two waves. So in theoretical problem, when the solution, when we deal with real equations, and we associate the two oscillators with the two equations, we can just define the coefficient of correlation in a standard manner, as in basic course of probability and statistics. If we compute these integrals, they are here just to recall the definition. Clearly, we are not going to integrate something, but if we compute, if indeed we perform the computation just with a pen or a pencil, we end up with a simple equation that the coefficient of correlation, the correlation between the two sine waves, is equal to the cosine of the phase difference. We are going to apply this exact formula, exact equation to approximate unknown phase difference, unknown phase difference phi, when we deal with observations, with two observed waves. So, we use these two curves as if they were the sine waves. They are not, but we pretend that they are and we compute the correlation between them. If you are not mute, you will answer, you will ask a question, what is the window where we compute the correlation? We will use the window that corresponds to the solar cycle. The fix some value, it's possible to fix 11 years, but here it is fixed to a bit lesser value, suggested by the state of the art. And the correlation, the computation of the correlation is performed within a sliding window of 10.75 years. Then we use the exact equation to estimate the phase difference. So we compute the arc cosine of the correlation, and we get the approximation of the phase difference in the Kuramoto equations. It means that we will know setter of t. Then what else? Are we just discussed formulation of the inverse problem of the Kuramoto model? We can still use that the left hand side of the second equation is approximately zero. We can get the approximation of the left hand side of the first equation. What is this? Let me remind that this is a proxy to the frequency of the solar cycle. So we can use the frequencies that corresponds to the solar cycle. And then we need to, I don't know how, but it's another question, we have to assign some values to omega and to delta omega and find k and delta k. Is it correct? Not exact. In theory, in theory, the two oscillators are synchronized. So the left hand side, the derivative of the phase difference is zero or approximately zero. And in theory, the phase difference is time independent. So this is the case with our approximation, because we deal with the data. We are able to compute the coefficient correlation for any signals, for any two signals. But the drawback of the story that we get time dependent phase difference. So it's not a drawback. In mathematics, we assume that the phase sum and the phase difference vary slowly. And what we obtain is called quasi synchronization, some generalization of the notion of synchronization. So to summarize this part, our data follow sign waves. The waves are shifted with respect to each other. And our assumptions hold to some extent about quasi synchronization. To some extent, because you remember about clear exceptions, we have already discussed at the very beginning of the data and during the 20s cycle. And let me stress that nowadays we see nothing special with two graphs, but we saw something, we saw some anomaly in instantaneous period in this anomaly underlies our main question of this part of the top. Let me return to the inverse problem. In our inverse problem, and we have instead of constant in the left hand side, we have time time dependent left hand sides of both equations, but we are able to estimate the left hand side of both equations through the computation of the correlation of the two waves. This gives us set of T and the derivative of set up and our knowledge of the system, physical knowledge, since we know that the period of solar cycle is approximately 11 years, we know the left hand side of the first equation. It's a good time to explain how can we estimate the frequencies set a one of T and set a two of T. In December, yearly I shown you the dynamics of the instantaneous period, and I told you that it is obtained it is derived from the instantaneous frequencies. Surely you know the mathematics related to this topic, and you know that the most popular way to obtain the instantaneous frequency deals with so called Hilbert transfer. If you Google instantaneous frequency, then your first references will be to Hilbert transfer, but this is not what I'm going to use. I will not tell you why the Hilbert transform is not the best tool in the case just, you know, just a hint. The Hilbert transform gives you information in a nice way about all frequencies related to the signal. In this particular problem, we are interested in the specific frequency, the frequency related to the solar rotation. This is exactly what we want to get. We want to derive, we want to derive the instantaneous solar rotation period, not all periods that we see simultaneously, not all regularities, but just solar rotation period. In computation, we use the Fourier transform. The Fourier transform gives us a lot of frequencies. So, we compute the Fourier transform over the solar rotation period T. So we deal with daily observations, and we give 11 year piece of time series as an input of the Fourier transform. What we obtained? We obtained first zero frequency with zero amplitude boost, but we can forget about it, for example, subtracting the mean of the data, and then the first coefficient is reduced to zero. And the next, the first coefficient corresponds to the solar rotation period corresponds to the frequency of the solar rotation period. We know this mathematics, so you have the Fourier expansion. You can make the first coefficient equal to zero, as I said subtracting the mean. But the point of interest is the first coefficient, zero coefficient, but the first coefficient. This is a complex number, and you can get the argument of this complex number and obtain the phase. And exactly this number is called instantaneous phase related to the solar rotation period. Okay. Anyway, I understand that this part is very, very technical, but as I said, I have a code for everybody who is interested and I can send this code. It's commented everywhere, and we will return to this to this part of this quote is a very, very end when you'll try to answer the question. We will return to the equations. As I said, I'm going to show the same equation. And as my lecture is going on, you remember this equation probably by heart and probably me too. As a result of the computation of the phases of the instantaneous phases, we know the green terms, all green terms are known, they are estimated with the data. Interested in the coupling in K and delta K, symmetrical and asymmetrical components that represent coupling between poor poloidal and to royal components of the solar magnetic field. Now what are omegas, what are natural frequencies. We have at least two possibilities to introduce omega and delta. At least two. First, we can guess affordable periods from oscillating data. We can use such omega and delta omega to that we mimic the range of observable periods. As I said, and you asked, and I answered that the period includes vary from cycle to cycle safe from nine to 12.5 years. We could define the average. The capital omega is average. In such a way that it corresponds to the average period to 11 years or to 10.75 it doesn't matter. Omega plus delta omega will correspond to 12.5 year periods and omega minus delta omega will corresponds to nine year periods and we get so in such a way omega and delta. By the way, to recall how to transform period frequencies. If the period is 11 years, then the frequency is approximately 0.57 years. The power minus one. And that is why we can choose omega and delta omega is in my yellow yellow highlight. But there is another possibility also very natural. We can use other factors. We can use the marginal, marginal flow. I explained you earlier, because exactly this mechanism transforms components of the magnetic field. And using known estimates of the speed of the marginal flow. We can roughly estimates omega and delta only. We will discuss the best choice or even the choice which is better than the best. At the very end, when we compare the results obtained with different values of omega and delta. But right now, let's use in some sense both possibilities shown here. And look at the result of modeling. So these graph have been already observed. This is the instantaneous period obtained from the data. Let me return back to the equation. I'm going to use this equation to find delta K and K. But then, when I find the coupling, I can use this equation to solve the direct problem to find. To find a phases and define the oscillators. So let me repeat. If I know delta K and K. Then apply the equations with known omega delta omega delta K and K to find theta and construct to virtual oscillators to model oscillators. The hope is that this model oscillators will follow the data. And this is indeed the case but these graphs are not in the presentation. But what is in the presentation is the instantaneous period computed with the model oscillators. Here is in the graph. So you see in blue. We see in blue, the data. And in green, the result of modeling. So if we compute the instantaneous period of model oscillators, we get, you know, we can go backward. I am very, very, I am very, very sorry, because it's, it's my, it's my fault. It's my fault. This is not true. I know that people like when lecturer does something wrong. And now we are here. And because I was wrong. In fact, the blue curve corresponds to the model and green curve corresponds to the data and spectacularly. The model curve is more smooth and this goes from theory it is correct. The curves agree with each other. But what is unexpected that it's the very end. The model curve goes exhibits extreme values animal exhibits anomaly with respect to the respect to the green curve everywhere. In previous anomalies of the green curve, the blue, the model curve looks like some kind of smoothing. It's not the case as the right, and we can ask whether it's a kind of model prediction. Surely we don't know it's only. It's only one way to look at the date, but we can. Remember it. Now, I'm going to show you the reconstructed coupling. We remember that there is symmetrical component K, which is in red and asymmetrical component delta K, which is the most interesting, most interesting thing here, and this is in blue. You can see the zero intervals of quite constant level of at least symmetrical component. But sometimes there are spectacular anti correlation between blue and red curves. The most famous and well observed interval is between 1960 and 19, I don't know probably 1975. This is the part where my mouse is, and this anti correlation caused by anomalous behavior of a index. You remember that at that time, a index failed to follow the source site. And you see that the chromata model exposed the full desynchronization between the two components of the solar magnetic field. It's not quite clear when we look at the as the data directly looked with an eight I receive this model as anti correlation between symmetrical and asymmetrical component of the solar magnetic field. Something similar. I was in the very beginning. You see the interval from the very beginning probably to the end of the 19th century with episodes of anti correlation. Let's look at the very end. And you see nothing. You see once more anomalous values of asymmetrical component values as such. But this is a single peak. And we see that the oscillation the variance of the blue curve quite large. So you cannot give a statistical proof that this last peak is significant. The question is, will we see the second peak as in the 1960s. Note that this picture is obtained with the average natural frequency implemented into the model equal to the frequency related to the solar rotation P. We use an alternative the alternative I announced earlier so different really very very different values of the natural frequencies. Up with quite similar pattern of the symmetrical and non-symmetrical coupling components. The tension that instead of a symmetrical component delta K, I placed some normalization, and this is a correct normalization that allows to compare that allows to compare on symmetrical component with different definitions of natural frequencies. The same pattern was typical oscillation about the same level during two episodes quite stable episodes and marvelous anti correlation between K and delta K during the 20s cycle. Some kind of anti correlation is the left and anomalous peak of normalized asymmetrical component of the coupling as a very well. So what I said summarized in this slide. So there are anomalies at the left. And they were caused, technically I mean caused by the absence of the constant phase shift between our proxies. Remember typically they were, but the loss of this phase shift at the beginning underlies this anti correlation at the left, I just mentioned. So then anomalous 20 cycle. When, when the two components of the solar magnetic field completely were completely desynchronized. And technically, K and delta K attains anomalous values as such, and they anticorrelate. And now nowadays asymmetrical component, asymmetrical component attains anomalous values, and probably by this speculation there are traces of the anti correlation between delta K and K. As I said, we apply some normalization of asymmetrical company. And if we draw the asymmetrical component obtained with different definitions of natural frequencies, we see their almost complete coincidence. So the question I raised, what is the best choice of natural frequencies is not a question at all. We can apply arbitrary reasoning, and you get the same result. We see how nicely agree, how nicely agree different graphs obtained with completely different natural frequencies. So this is not something we are going to discuss. Next, with current slide, not next with current slide. I'm going to have a look at the future, and to make a step to solve the enigma of nowadays quasi anomaly, anomaly or not. I remember that my previous estimates, previous reconstructions of the coupling are smooth. Why they are smooth? Because when I use model equations, I have to find the derivatives of the curves. The derivative is, you know, is the difference between two neighboring values of the function. But if you indeed apply such a natural definition of the derivative, you get very unstable results. As a smooth picture, instead of derivatives, I use very many sequential points, four years of points, and the derivatives were computed as a trend, a four year trend. Then the picture is quite nuts. Turning from four years to one year trends in the computational derivatives, I get this not so smooth picture, but still you see all the details we discussed earlier, for example, anti correlation in the 20th cycle. But now we get some more points at the end. And you see that in addition to this peak of blue curve, this blue curve goes upward at the end, but that's it. It can go farther, probably some more points, but not the peak or the absence of the peak. We are waiting for another portion of data, probably two to three years. And with this method, we can get an answer. Probably other methods will lead us to the answer area. So this is some kind of competition. But if not, we will answer in two years with cool that is ready to be shared. Whether we are experienced the epoch of complete desynchronization between toroidal and collo and colloidal component of the solar magnetic field. Now, just to conclude, the current model successfully reproduces the variation of the solar cycle duration. It describes the loss of the synchronization in the 20th cycle, and uncovers traces of the current synchronization, but requires additional data for a definite conclusion. And as I said, if you want to look at vital code, which is, you know, very convenient to follow at home because at each step you have a lot of pictures, a lot of additional pictures. And so it's quite clear quite nicely commented on, please write me in the mail, and I will happy to discuss any part of the story with any of you. Thank you very much. Thank you very much for the really exciting lecture in terms of understanding really how I said mathematicians or how theoretical geophysicist physicists are using the simple models like a model to promote or can explain such a complicated problems.