 We have, as you know, it's something, 45 minutes, and then it is a 10-minute break, and then another 15 minutes. Thank you. Thank you very much for this introduction. Let me express my deepest gratitude to all the organizers who invited me to this Marvellous event. And it's a honor to have a floor, or I don't know, a screen here. You know, I'm, as Ali just said, in a cold Moscow, and the workshop is in Trieste. So I can imagine, see the sun. By the way, the sun is the subject, is a topic of my today's talk. My plan is quite simple. First, I'll introduce a brief, but I hope exciting description of the sun as a complex system. And then I'll take an advantage of this full lecture, and not just 20-minute talk, to give you an example of a solved problem, a problem of sudden desynchronization of the components of the solar magnetic field. I'll show you the complete way from the formulation of the problem to a possible solution. So we'll discuss even some technical details. I have a computer code, which I cannot share in the format of the lecture, because it is impossible to follow the code. But the code is written for everybody with full explanations. So I can send it to anybody who is interested in. The mathematical component deals with the so-called Kuramoto model of cow-pulled oscillators. And we discuss the simplest case when there are just two differential equations, but not independent. They are cow-pulled. And we will see what is this and how to use this model, these two equations. So the first part, the sun is a complex system. You will see a lot of pictures. And so this part, I hope, is really very exciting, because we can see the sun from different points of view. First of all, the first picture is the sun itself in different scales. So at the left, you see the sun and some black points, black areas. And I know, look at it at the picture. These black areas are called sunspots. They are visible with a telescope. And these black areas, these sunspots is an input of many, many models, looking at them, observing them, processing them, collecting information, and, in fact, can reveal a lot about the sun. But just now, using these marvelous pictures from the NASA websites, both of them show us these sunspots. And you see that the sun is extremely heterogeneous. The sun gives us also an example of regularity breaking. Because the sun is rotating as the Earth does. But you know the period of rotation of the Earth. It's one day and day, night, day, night. So we are accustomed to this rotation. At the first sight, it is difficult to believe that the sun exhibits differential rotation. The period of the rotation depends on the latitude. And you see the picture, which represents different periods, as I said, depending on the latitude. The period varies from 25 to 32 days, depending on the location. And the average period, the number that you can see everywhere, it is approximately 27 days, as it is observed from the Earth. Let us return to the sunspots, which will be the main character of my today's presentation. And return to heterogeneity, heterogeneous observations. Because the sun is an example of heterogeneous observations. Now you see the sun at, I don't remember when. And you see some black areas. And now you know that these areas represent sunspots. And some of them are clustered. Scientists call each cluster as a group of sunspots, or sunspot group. And you see two sunspot groups in this picture. But two and not two. Probably three to four weeks later, the sun is a little bit different. The activity increases. And as a result, the number of the groups, not two, but four now. And you see that these black areas are larger than at the left. And these areas, these sunspots, generate a solar index, index of solar activity in early days called the wolf number after wolf who proposed this index. It is computed. It is defined as the number of groups four at the right multiplied by 10 plus the number of sunspots and multiplied by a specific coefficient, so-called observer coefficient. We will look at this proxy at wolf numbers as a proxy to solar activity many times during today's lecture. By the way, the name, which was the wolf numbers, changed. And now scientists address to it as international sunspot numbers, ISSN, this notation will appear later. But now let me give a historical note. It was Ludovic XIV, the sun king, Rudi Soleil, who declared to observe sunspots. And what happened at the time, almost immediately after the sun king, do not forget the sun king, declared to observe them, they disappeared. They disappeared. And the poh, called the Mounder minimum, or Mounder pose, started. It's really very interesting that, at some moment, solar activity was very, very small. It lasted for almost 100 years. And to see it, you have to look at this graph, in this graph, at the red curve, which is almost invisible, because it almost coincides with the horizontal axis. This is the level of activity. So it dropped in the beginning of 70th century, almost to zero. And some values were recovered only in the end of the century. You can say, well, it was closed because no one observed, even despite the king declared to observe. But it is not the case, because there were observations. And the number of observations is shown as a background. So we see that people, a lot of people, here is a number of observations that was made, that is in blue. And despite these observations, almost nothing was observed. So you can look at the sun as an example of numerous observations. The sun is an example of cyclicity. And it's not just a picture of a butterfly. This is another thing. Each vertical line represents a lot of dots. And these dots are observed sunspots. Sunspots observed along, this is the misprint, along the logit, logit. So the vertical axis represents the location along the logit. And for example, as the very left, the sunspots were observed, well, somewhere in the middle latitudes. Here, I hope that my mouse is visible. The sunspots were observed close to the equator. At some time, because the horizontal axis represents time. So some time, as for instance, I don't know, but for instance, here where is my mouse, the sunspots were observed in order, let me say, like this. So let me collect many regularities of this picture. Well, first of all, it is Sasha. You know, we have a problem with your lecture because it's interrupts. Maybe you have a low internet activity right now at your place. But is it possible to switch off your video? Now you are muted. Please unmute. Yes, but still I have a problem. I don't hear you. I don't know others. Could you write in the chat? Do you hear Professor Shupaval? OK, now unmute please yourself. I'm sorry for this. Yes, now it's OK. Fine. It's very strange because I understood the problem. But all the time, well, OK, the connection is nice. But anyway, now it's fine. Yes, now it's fine. Let's follow. So let me share the screen. Wait for a moment. I'll have a look at chat one moment. So my plan is to start with this slide. So let me go backward. Previously, I discussed this picture and tried to say that the sun is an example of numerous observations. And I stressed your retention on the mountain pose when the sunspots was almost absent. And it is shown by the red cube. And now I'm going to show you the proposal to look at the sun as an example of cyclicity. What is here? The horizontal axis is time. The vertical axis is the location of sunspots at a given day along the longitude. This is a misprint. It is not a latitude. It must be longitude. And you see that at different time moments, sunspots are located at different latitudes. For example, they are almost never located at high latitudes close to the poles. Then you see the equator world drift of sunspots as the cycle proceeds. Let me recall that from the left to the right, the time goes on. And you see the equator world shift from left to right. You also see that, well, at the first sight, that the picture is symmetrical with respect to the solar equator. But sometimes, for example, here, where my mouse is, you see a clear asymmetry. So we could expect the symmetry with respect to the equator. Still, the astrophysicist is interested in symmetry breaking. The goal is to explain why the symmetry is broken from time to time. And we will return to, well, reasons. Let me say reasoned episode of asymmetry, which happened in 1960s to 1970s in so-called 20s cycle. And interestingly, that frequently at solar minima, the sunspots of both cycles are visible. Solar minima means that the solar activity is in its minimum. And during the minima, the sunspots of both cycles are visible. Sports of a new cycle appear far from the equator, like here. And at the same time, you see the sunspots close to the equator. And they are the sunspots of the ending cycle. And, well, you can guess the name of this picture. It is called butterfly diagram, because it reminds us a butterfly. I talked about cyclicity. And you see this quasi-cycle from minima to minima, from minima to minima. And it's approximately 11 years. It's called solar cycle. It lasts approximately 11 years. And this is, let me say, the main regularity of solar activity. But because of change of polarity, the magnetic cycle consists of two sequential solar cycles. And that is why we have another regularity related to the solar magnetic field, which is approximately two solar cycles or 22 years. It is called hail cycle. So let's return to wolf numbers that were improved, let me say, like this, by different teams. And now they are called international sunspot numbers. They are observed daily from more than 100 years, from 1870, we have regular observations without gaps. As you remember, these observations started at the time of Ludovic of Louis XIV. And from 1870, we have daily observation without gaps. And you see these daily observations, a number of cycles. And you can guess where are minimized maxima. And we can smooth the data with four-year moving window. And then you see a wave. And it's a good question. How similar is this wave to a sine wave? And we will return to this question later. And 11-year cycle is quite visible. But let me stress that it's not a cycle. It's not the exact regularity. It's quasi-regularity. It's quasi-cycle because the period in quotes and amplitude vary from cycle to cycle. And you can even see that from the beginning of the 20th century, the cycles are in general larger. So we see a long wave. And this is indeed another regularity called Gleisberg cycle. Cycles, it lasts. Well, it's called secular cycle. The length is not extremely clear. It's like four sequential solar cycles. So up to 8 multiplied by 11 is 88. So it's up to 100 years. Now we can look at solar cycle and other cycles with different scale. Now you see the observations, the proxy of solar activity, another proxy group number. And I explained what is the group of sunspots. And you see this index from the time of Louis XIV. And here, in addition to the solar cycle, which is clearly observed, we see the Mounder minimum or the Mounder pose at the left, another minimum not so strong, which is called Dalton minimum. And you can guess this large secular wave. And this minimum is a part of this wave, which is so this minimum is extremely deep. But you can believe in this large wave of approximately 100 years old Gleisberg cycle. If you look further in the past, you can see even larger cycles. We know cycles, quasi cycles that last for millennium, cycles that last for approximately 2,000 years called Holstadt cycle. And you see that it looks like you have periods. The periods are doubled from 11 to 24, 22 years, from 22 to 88, eight cycles in a row, from 1,000 years to 2,000 years. And you know that this is a trace of low dimensional chaos. So you can treat the sun as an example of a low dimensional chaos. It's not a single trace, which I just shown you. There are many related to well-known characteristics of chaotic systems like Laponoff exponent or fractal dimension of related proxies. But just to give you intuition, I show this simple picture. And I'm not going to go beyond that. Again, just to give you an example of synchronization, which is observed with solar proxies. And now we can look at the sun as a system with the sun as an example of synchronization. I presented a lot of different time series related to solar activity. And I'm not going to quantify them right now. But you see one, two, three, five different series. They are, their definition are absolutely different. But because they are related to the sun, they are related to one another, and they follow one another. You see almost, well, you see very similar pattern exhibited by each of these series. So the sun is an example of synchronization. Now I want to say that the sun is an example of de-synchronization. And as a first sight, well, it looks that I'm cheating because these two series are nicely synchronized. You see almost exact anti-correlation, almost exact anti-correlation. So you look, it's almost as if it were a mirror between two series as the scale which is presented. But if we change the scale, if we consider the scale of several years, you can see, and I will show where, these two nicely synchronized proxies suddenly exhibited de-synchronization. For example, last points, it's very right. You see that the both curves go downward, go to the same direction. And this is not a single anomaly. Let's look at the minima of the lower curve and the oscillation around some level between 1970 and almost 1980 where this anti-correlation suddenly disappeared. And as a result, you see the change in the scale. You see that even nicely synchronized series could lose de-synchronization. You see an example of sudden de-synchronization, example of anomalies that have to be explained and understood. And the sun is an example of one of those predictions. And this is very interesting and exciting and intriguing picture. Well, if you know a lot about a system, you could try to predict its future. It's very interesting if you think about the sun. Regarding the solar cycle, you could try to predict the time of the next solar maximum and the amplitude at the maximum. This picture summarizes a lot of predictions of the maximal amplitude of the previous cycle, cycle 24. Now we are in cycle 25. The predictions are summarized according to their methods. So each rectangle represents a range of predictions, roughly speaking, done with the same method. And black dots show you the average obtained with each method. The area shows the variation of the prediction, the variation of this black dot, if you allow me such speaking. The black dashed line is the global average, the average of all these predictions. The blue line is the real value. And you see that the majority of predictions are larger than the true value. So we can conclude that even despite the scientists know a lot about the sun and are raised successful in understanding the solar dynamo mechanism, we have a lot of work. So we should not avoid working with the solar data because everything is known. No, not everything. Our predictions can be significantly improved. To conclude this part, I would say that the sun is a simple, natural example of a complex system with continuous development of knowledge, huge data volume, non-linear modeling, and non-trivial prediction. Well, the part with a lot of pictures is over. And now I am going to introduce an example of an inverse problem when we will try to identify of episodes of desynchronization. To start, I recall you the butterfly diagram. And you remember this marvelous story about equator word drift of sunspots as solar cycles proceed. And now let's think about some explanation about physical mechanism that stays, that underlies this drift. First of all, we can think that there are different components of solar magnetic field. There is so-called toroidal component of the solar magnetic field related to the equator. There is a poloidal component related to the poles. And magnetic field performs a transformation. I answer if it leads, if you don't mind. And so the last, the minimum was OK. So there is a question, you said, OK. You said amplitude vary from cycle to cycle, which is very clear. Can we also call that solar activity level? I'm not sure that I understand this question. I think that during the question and answer session, you'd better explain a bit more. But the last, the minimum was in the beginning of 2009. It's regarding your second question in the chat. So let me now return to toroidal and poloidal components of the solar magnetic field and describe very roughly, very, very roughly, the transformation between how magnetic field transforms from toroidal magnetic field to poloidal and then back. You can think about a flow, a flow which is sketched in white in this figure. So it's done with my hand, with my pen. And this brief and rough presentation, to some extent, correct. Indeed, the transformation from toroidal to poloidal components is performed in the solar surface. And the backward transformation happens in deeper layers of the sun. Recently, some observations with satellites become possible. And we know the speed. We better know the speed in the upper layers than in deeper layers of the sun. But still, there are estimates of booths. And this part of the flow is characterized by larger speed. And the motion is very, very slow in deeper layers. In what follows, we will need proxies to both components of the solar magnetic field. And we have them. One of the proxies is sunspot numbers, wolf numbers, international sunspot numbers. So the series I have already shown. The second proxy is called magnetic AA index. And it corresponds to the poloidal component. What is important, now we see the picture, the graph of the two proxies, where sunspot numbers, ISSN, is in red. And the second index is in blue. These two smoothed indices exhibit a phase shift almost everywhere. So we see that the blue curve go a little bit ahead. Not everywhere. You see clear exception at the beginning, at the very beginning, then, but probably it's relatively short period of time, but still, then, as the second, full maximum. And this phase of synchronization lasts probably for a cycle up to the next, almost to the next maximum. And then blue curve goes ahead. And in the 1960s, you see another moment, first of the synchronization of the both indices. And then, even a disaster, because the blue curve for a moment stopped following a solar cycle. So let me repeat. The red curve, which is quite smooth, follow the solar cycle everywhere. It's not surprising, because this is a proxy to solar activity. And you see this wave everywhere. It's not the case for geomagnetic index in the 1960s, because you see, this is a wave, then it goes down, then up. And instead of going down after a short stop, it continues upward moving. So something happened during the 20s cycle, during the 1960s. And it is, you can see it with innate eye. And there was, and probably still is, a discussion of what happened at that time. And let me stress your attention that nowadays, you do not see something like 100 years ago, something like in the beginning of the 20th century, or in the 1960s. I need a minute or two to pose the problem, and then we'll make a break, 10 minute break. What I'm going to show you is a story about the period of solar cycle. We understood, it's clear, that it varies from cycle to cycle. We can, and probably this is another attempt to answer the question from the chart, we can give value. We can compute the period. We can compute the instantaneous period of solar cycle using Fourier transform. I'll postpone details to the second part, how to compute. So let's assume that we are able to perform the computation. And the instantaneous period computed with ISSN is in orange, with AA in blue. And the average, let's also postpone what doesn't mean the average in green. And you see the oscillation around 11 years, which is expected because the period is approximately 11 years. But then you see at least two exceptions. In the beginning of the century, in the 1970s, it's the peak, so in the 1960s and 1970s. And it's the very end, which corresponds to nowadays. And let me recall that with this graph, we see nothing interesting at the right. And the main question is, will anomalous growth in the instantaneous period results in desynchronization of toroidal and poloidal components of the solar magnetic field? I am going to start my second part exactly with this slide. And probably, first of all, I answer questions. And now, Alec, my part, my 45 minutes are over. Yes, fine. Yeah, thank you very much, Alexander. It actually is a fantastic knowledge, at least for me, about the desynchronization and desynchronization. Indeed, we are talking about the complex Earth, but some, I think, will give us more and more surprises with time of the gathering information and data from this. And indeed, the socialization of this data and analysis of this data and the understanding through the inverse problem, it's very important. Thank you very much. Now, 10 minutes break. And then we will.