 Welcome to this session of seminar. The first speaker this morning is Professor Ilyashenko, he's Professor Moscow and in Cornell, and he will speak on limit cycle from Poincaré to nowadays. The problem given F to find the functions fx of t that satisfy this equation. On the other hand, they are subject of geometry. At any point in the xt plane, we have a direction, a straight line with the slope given by this function. And an equivalent problem is to plot the curves that are tangent to these directions. This geometrical interpretation was well known in the time of Poincaré, but Poincaré was the first to say that differential equations are in fact a branch of geometry. At his time, it was already well known that for majority of the right-hand sides, the explicit formulas for the solutions simply cannot be found. And therefore, Poincaré said that the differential equations should be studied by the right-hand sides directly. The picture like this is called phase portrait, splitting of the plane to the orbits of the differential equation. And Poincaré was the first to plot the phase portraits of differential equations near the singular points. This is, if f is, say, a ratio of two polynomials, a singular point is a common zero of these polynomials. These are the famous pictures that Poincaré obtained. They are called saddle node, focus and center, following Poincaré. And together with the geometric theory of normal forms, Poincaré established, sorry, the geometric theory of differential equations. Poincaré established the theory of normal forms. The paradigm is the following. We cannot solve the equation well, but probably we can find a coordinate change in such a way that the equation will be drastically simplified and in new coordinates it will become solvable. For instance, the previous pictures were plotted in bad coordinates. If you find good coordinates, they become linear, and you may solve the equation and plot the explicit phase portraits. There is a deep theory behind this statement. And Poincaré himself has found an appropriate coordinate change not for all the differential equations in the collection above, but for so-called non-resonant, whatever it means, nodes, and fozzi. These phase portraits, after an analytic change of coordinates, become linear and maybe easily studied. Only 60 years later, Ziegel found an appropriate statement of the theorem about the linearization of a saddle and what is a normal form for a center was understood only in the 80s in the last century. Closed orbits on the plane of such equations are cycles. If there are no other cycles in the vicinity of the orbit, they are called limit cycles. And Poincaré was the first to introduce this geometrical object and he said that limit cycles like torches enlightened the way to the understanding of the phase portraits of the differential equations. At this spot I should confess that I quote Poincaré not from his texts but from the quotations by other people. So if I should be a historian, I should write attributed to Poincaré at this spot, but allow me to be not too much accurate. Here is a limit cycle and Poincaré introduced a very simple map yet not considered beforehand. You take a cross section, you consider orbits emerges from the points of this cross section and the point of the first return. You get a map, a self-map of an interval into itself and you may say very much about the behavior of the solutions of the differential equation near the limit cycle. Just studying this first return Poincaré map. For instance, if you have other limit cycles nearby I did not plot them. Then you will have another fixed point of the Poincaré map. If the map is analytic then this is the graph of the Poincaré map. This is identity and this is the graph itself. If the vector field is analytic, if the differential equation is analytic then the Poincaré map also is analytic and by classical theorems from complex analysis it may have only a finite number of fixed points. So as a trivial consequence of the very concept of the Poincaré mapping one may say that for an analytic vector field I will show this picture later limit cycles cannot accumulate to a closed orbit. As limit cycles are very important then sort of a hunt for limit cycles makes sense. And Poincaré made the first steps in this direction. He considered, let us first look at the left domain. There is a domain and an analysis in which the vector field in the plane is pointing inward. And no singular points are allowed in this domain. This is the data. What is inside is obscure at the first glance. Poincaré claims that there is a closed orbit inside such a domain. If the vector field is analytic then this closed orbit is... there is a limit cycle inside this domain. This theorem is called Poincaré-Bindixon theorem. For a long time I thought that Ivor-Bindixon was a student of Poincaré. Not at all. He was an independent mathematician brought up in Sweden and he just gave a rigorous proof of the previous theorem that Poincaré simply sketched. Bindixon in 1901 wrote a memoir about the differential equations and it was called precisely in the same way as the famous memoir by Poincaré sur le courbe define par le casier aux différences. In his memoir there are several chapters. The first one is dedicated to the Poincaré-Bindixon theorem and the other four or five chapters are dedicated to a theorem which was born but not became completely mature in the Bindixon's memoir. A lot of authors continued it and they are listed here and the Bindixon's theorem claims that however complicated the limit point might... the singular point might be after a finite number of so-called desingularizations it may be split to simpler ones. First of all you should require that the differential equation is analytic and the singular point is complex isolated. Secondly, simpler singular points are from the previous list of settles, nodes, force centers and one extra example should be added. It is so-called saddle node. The example of the differential equation is here and all these points are elementary singular points. You take a complicated singular point, you may a polar blow up, you pass to polar coordinates, you paste in a circle instead of the singular point and you get a simpler picture. In my example the simpler picture, the elementary singular points appear in one time and when you have this picture it is easy to study the original one via the projection. Another quotation from Poincare that I like much is that the truth is born as a paradox and dies as a triviality. The Poincare, the desingularization theorem was not proved in the memoir by Bindixon and first several people gave a complete proof and then Van der Neissen gave a simple proof and only at the beginning of this century the Bindixon theorem, the desingularization theorem was included in the textbooks and at this spot we may say that it becomes sort of a triviality. Now let us pass to the major problem in the field to the Hilbert 16th problem. In the year 1900 Hilbert stated his famous mathematical problems and the second part of the Hilbert 16th problem what may be said about the number and location of limit cycles of a polynomial vector field of degree n. In this interpretation f is a ratio of two polynomials of degree n and we ask what is the maximal number, for instance, of limit cycles. Or another form, another specification of this question is it true that the number of limit cycles for a polynomial differential equation is finite. The maximal number of limit cycles is called the Hilbert number. It is denoted by h of n and nobody knows yet whether this number exists. Even now we do not know whether h of 2 exists. So the Hilbert 16th problem second part is one of the most persistent problems in the famous Hilbert's list. In 1923 Dulac published a large memoir where he stated the following theorem a polynomial vector field may have but a finite number of limit cycles. The first step of the proof is to start with a contradiction, with a contraposition. Suppose that there is an infinite number of limit cycles. A polynomial vector field may be extended to the real projective plane before the situation is compact and these limit cycles have to accumulate somewhere. They cannot accumulate to a closed orbit by analyticity of the Poincaré map and this is a simple argument that I explained. Now one should prove that they cannot accumulate a so-called polycycle, a separatrix polygon. You have singular points and the solutions that together form an analytic curve through the point they are called separatrices. So this polygon is formed by singular points and they are separatrices. And one should prove that the limit cycles cannot accumulate to a separatrix polygon. Dulac's proof has three steps. The first step is using the desingularization theorem by Bendixon. At that time it was quite a non-trivial fact and Dulac paid much attention to it. Then Dulac studies so-called Dulac's maps that I will show in the next picture and then studies their compositions. The Dulac's map occurs only near saddles and saddle nodes. The geometric picture is almost similar but the simplest example for the saddle is a power Dulac map, x comes to x to the lambda and for the saddle node the simplest example is exponential. x comes to the exponential of negative 1 over x. So-called flat map. All the derivatives at 0 are 0. So Dulac studied these maps. Ignore the lower lines right now. For a while I stopped with the Dulac's problem and switched to the applications of limit cycles. I want to tell you a few words about Russian mathematicians of the generation born at the very beginning of the 20th century and one of them is Russian mathematician and physicist Alexander Andronov who was one of the first to understand that auto oscillations in physical devices are modeled by limit cycles. So you have some physical device, radio device for instance and it works in a stable regime. The external power for instance is constant and nevertheless oscillations occur inside. What is the mathematical model for that? Andronov understood that the mathematical model is the Poincare limit cycle. Andronov have done two great things besides. First he decided to work not in one of the major centres, one of the capitals in Russia but rather to create a mathematical school in another city, Nizhny Novgorod and the mathematical school is now very strong and well known. Also he was one of the founding fathers of the bifurcation theory. For instance he was the first to investigate the appearance of a limit cycle from a singular point and when the parameter in the family that he investigated achieved a critical value and the singular point was ready to generate a limit cycle Andronov said that the singular point is pregnant by a limit cycle. Another person whom I want to mention is Ivan Petrovsky who was a rector of Moscow State University in the last 22 years of his life and who attacked both parts of the Hilbert XVI problem. I mentioned 10,000 of good deeds that he have done. This is an estimate of one of the scientists of his generation. The fact is that in his time there were two parts of the scientific community. One was, as one may say, ideologically faithful and another was talented. There was an intersection but it was not too big and Petrovsky was a center in the Moscow University of the second community and these thousands of good deeds are lives of talented people who were not ideologically faithful but whom he allowed to realize themselves and to work in the academic world. A few words about the first part of the Hilbert XVI problem. This first part deals with something also algebraic. This is now not a differential equation with a polynomial right hand side but rather a polynomial itself and its level curves. The first unsolved question in the Hilbert's time in this domain was the following. Consider a polynomial of degree 6. It was well known that it cannot have more than 12 closed components of the level curve. This is one level curve of the polynomial. 12 ovals in total, one outside and 11 inside. Stating his problem, Hilbert claimed that he can prove that the picture that I showed here is impossible. But the first rigorous proof was given by Petrovsky in 1938. The tool that Petrovsky was, was the following. Complexify the problem. He extended the problem to the complex domain. He worked in the complex domain and then he got an answer for the real problem. Together with his young student Evgeny Landis, Petrovsky enterpised an attack to the second part of the Hilbert's problem. The strategy was the same. To complexify the problem, to study the problem in the complex domain and then to get an answer for the real case. Landis at that time returned up to Moscow State University after six years of war. He was a soldier for six years. He miraculously came back alive and he immediately started the scientific work even as the student of the first year. Together with Petrovsky, they succeeded to make first steps in the theory of complex polynomial fluctuations. But their result, published in 1955, was disproved in 1963. The following very interesting persistence problem was claimed to be solved in their paper. But it was not in fact solved and it is not solved even up to now. The question is the following. Suppose that the right hand side of the equation depends on some parameters and the equation has a limit cycle. We begin to change the parameters. It is very easy to construct a situation when the limit cycle will disappear. It happens like with the roots of the square polynomial of the quadratic polynomial of the square equation. They may pass to the complex domain. But the limit cycle disappears from the real plane yet preserves in the complex domain. So the question is, is there a domain in the parameter space through the boundary of which it is impossible to extend the complex limit cycle when you change the parameters? The answer is still unknown. And this was a gap in the Petrovsky-Landes solution. In 1981, as well, the du Lac's result was disproved. Freddie du Mortier giving lectures in Brazil in 1977 claimed that he doubts whether the proof is complete. Robert Mussu in 1981 sent a letter with a question, colleagues, what do you think on the du Lac's proof? And at that time, I was ready to point the fatal error in the du Lac's investigations. Let me explain in a few words where the error was. As you remember, there were three parts. The singularization, which is okay and even simplified at the new time. Study of du Lac's map for settles and saddle nodes. Du Lac made important progress in this study. But his study was in the C infinite category instead of analytic. And this was insufficient. And in the study of the compositions, du Lac made a fatal error. In this picture, I present a schematic history of our knowledge about limit cycles. This is not a face portrait, yet this is a time axis. And this graph is the rate of our knowledge about limit cycles and the Hilbert-16 problem. P is Poincare, H is Hilbert, D is the du Lac's theorem of 23. P L is Petrovsky-Landes, the theorems of 55 and 57 when we thought that the 16th Hilbert problem is solved. Then after the disproval of their work, we have a fall down. And we stay on this level, then we have another fall down in 81. And we are on the level approximately the same as at the beginning. Then there was some accumulation of information. Yet there are some branches that grew up, which are normal forms. This is not an institute, this is infinitesimal Hilbert problem. This is analytic fallations, resurgent functions by Eccl. These are bifurcation, these are functional moduli, these are nonlinear stocks phenomena. These are restricted version of the Hilbert's problem. Some way up was due to Eccl and myself when the du Lac's problem was solved. So a few words about these developments, probably I will skip some. I will probably skip the infinitesimal Hilbert problem in details. The problem is the following. We have an integrable polynomial differential equation epsilon equals zero. And we add a polynomial differential form or a polynomial vector field multiplied by epsilon. This equation is integrable, all the solutions are form families of closed curves, different families. And the question is how many limit cycles may be generated under this perturbation? The answer is related to the number of zeros of this integral. It is the infinitesimal Hilbert problem. Find an explicit bound for this number. And this problem was very recently solved by Yakovian-Kanovykov and Benjaminin. So we have sort of a first number in the theory of limit cycles. You have some problem about polynomial differential equations with the restriction to the degree. And you have some upper bound related to the number of limit cycles. Now there is another problem that may be called Hilbert-Arnold problem. This is the following problem. You take a separatrix polygon, a polycycle, and you perturb it. The question is how many limit cycles may occur? The problem is stated not in the polynomial context, but rather in a new context related with the philosophy of general position. You take a generic family of smooth vector fields. Instead of the degree of the polynomials, you consider the number of the parameters. The larger is the number of the parameters, the more complicated is the polycycle that may occur in a typical k-parameter family. And the question is how many limit cycles may be generated from this polycycle? It appears that if the parameter is, if the polycycle has only elementary singular points, then the answer may be given. And the final result is by Vadim Kaloshin with this explicit estimate. It is another explicit number related to the problem of the number of limit cycles, once again in a particular problem. There is an intensive program developed almost 20 years. From now, Demortier, Sari and Rousseau tried to prove the existence of H of 2. The existence of this number would follow from the following statement. Any polycycle, any separatrix polygon that occurs in the family of quadratic vector fields, that is vector fields with degree 2 polynomials in the right hand side, some polycycles may occur in this family. If you prove that any of them may generate only a finite number of limit cycles, then by simple arguments you are done and the existence of H of 2 will be proved. So, they listed 121 different classes of polycycles and in the past almost 20 years, more than 80 of them were resolved but about 30 are not yet studied. I will show you one very, very simple, probably not too much expected picture that will show you what specific cases are not yet studied. You can easily write a linear vector field whose face curves are circles. You just erase this bracket and this is a very simple classical equation. Then you multiply this linear vector field by a linear function and you get a straight line of singular points. This is a quadratic vector field. You can very easily plot the face portrait, no problem about that. But any of the closed continuums here, this one with this red singular point, this one arc of a circle in the segment of singular points and there is a continuum of such sets. Any set of this kind may generate a limit cycle under perturbation and the finiteness of the number of limit cycles generated is not proved in this case. There is another branch on the picture that I have shown, complex polynomial fluctuations and I only want to stress that the persistence problem for complex limit cycles which is in this domain is still unsolved. There are some important news in the theory of normal forms. Nonlinear stokes phenomena is related to this and it is mostly related to so-called parabolic fixed points. We start with a conformal map like this by a formal coordinate change without taking care about the convergence of series. We can reduce it to this normal form but the normalizing series are divergent and a new class of normal forms appears. Namely, the normalizing series is asymptotic to a holomorphic map which has singularity at zero. It is defined in a sector with the singular point on a boundary. In this sector, the map may be transformed to the normal form. In another sector, it also may be transformed to the normal form and this couple of normalizing maps is called normalizing co-chain and this is an important new element in the theory. The moduli of analytic classification are the transition functions between these two maps. They are a Calvaronian moduli. And normalizing co-chains appear in the study of saddle nodes produced by Martinet and Ramiz. And I show you once again the same picture which Dulac investigated. The analytic version of the description of the Dulac map for the saddles brings us to so-called almost regular germs, some special class of analytic maps. And the study of the Dulac map for saddle nodes brings us to functional co-chains that I have described right now. In the beginning of 90s, the following finite terms for limit cycles were proved. A polynomial vector field in the plane have but a finite number of limit cycles, exactly what Dulac claimed. And limit cycles of an analytic vector field in the plane cannot accumulate to a polycycle of this field. Once again, let us return to three steps that Dulac have made. The singularization, it stays as it was and is even simplified. The description of the Dulac map requires quite new tools and these tools were developed in the 80s by the people whom I list here. And the last step is composition of the Dulac maps. It is a very difficult part and it is mostly the problem to be solved in the proof. Let us say a few words about further results. Murtada published a preprint, not yet published in any journal, with the statement that a hyperbolic polycycle of an analytic vector field may generate but a finite number of limit cycles via a perturbation in the domain of analytic vector fields. So hyperbolicity means that all the singular points are hyperbolic settles on this polycycle. Probably this is one of the major statements claimed. And I want to conclude with pointing the major open problems that remain in the field. Is it correct that the Hilbert number exists? That is, is it correct that there is a uniform upper bound for the number of limit cycles for any polynomial vector field of degree n in the plane? The upper bound should depend on n only. The question is still open, even for n equal to as I said. Another more difficult problem is everybody believes that this upper bound exists. Give this upper bound. Smale simplified this problem, these two problems, and he suggested to consider the family of Lienar equations. A lot of questions are much more simple for this equation than for the general polynomial equation. The Lienar equation has this form. Only one polynomial of one variable is in the right-hand side. Still we do not know the answer. So at this spot I stop. Thank you. Thank you very much, Professor Yasheng Fu for wonderful lecture. Are there any questions, comments? Thank you. I have a question and a comment, both historical. First you said that Andronov was a founder of bifurcation theory. I would rather say that it's Poincaré. Already in his thesis in 1879, there is a proof, I think, independent of Viastras of the Viastras preparation theorem, which is fundamental in bifurcation theory, and there are many studies of bifurcation in Poincaré's celestial mechanics, for example, works. The question is, I thought, but I never saw the picture that Zhukovsky, before Poincaré, had drawn the pictures of saddles and foci and so on. So maybe you can comment on this. I never heard about the pictures drawn by Zhukovsky and I did not compare the times. Thank you. About the return map, Poincaré certainly was knowing the Kronecker index and Kronecker used, before Poincaré in some sense, the Poincaré map from his index. Jean-Marie is recalling me that Jean-Marie Ginou from Toulon found unpublished, at least unknown, conferences of Poincaré. I don't remember some school of electricity, I don't remember exactly, on the use of limit cycles in electricity and so on, which were earlier than the application you mentioned also. And his thesis is about this question. Arnold liked to say that the school of Mandelstam and Andronov did not use what Poincaré had done, so it is parallel to what you say. Could you come back to the finiteness theorems for... I'm here. Could you come back to the finiteness theorems for HN? I didn't understand why they don't apply the existence of HN. I must have missed some steps. What place do you... The finiteness theorems that were towards the end. The statement of the finiteness theorem, they are at the very end. Here they are. Is it what you want to see? Yes. So these theorems are proven? Yes. But then... Oh, no, okay, I get it. Thank you. There is a book published by École in 92, and another one by myself in 91. Okay, thank you. Another question? Yes, so to add a comment to what I've said, so it's in some lectures of Poincaré at the École Supérieure, they post a telegraph that in one of his chapters, when he was dealing with radio problem, Poincaré explicitly connected the singing arc coupled to an oscillating electrical system which was used before the triode in radio. He connected it to a second-order equation which is essentially of Lienard type, but without explicitly the structure of the nonlinear part. He kept it. So he had made the link between his limit cycle concept and the self-association in a radio system. Already, I think it is in 1908. This has been recently discovered. Unfortunately, in contrast with the other lectures of Poincaré at this École de Petite, this is not included in his complete work, which are so less completed than expected. Well, I guess that Russian school was not well aware of what was going on. Yeah, but it's normal. Nobody was aware. Yeah. Other questions or comments? Also, we thank you once more very much. Thank you.