Russian Mathematical Surveys, 2025, Volume 80, Issue 2, Pages 345–357 DOI: https://doi.org/10.4213/rm10227e (Mi rm10227)

DOI: https://doi.org/10.4213/rm10227e

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Generous sharing of results is a tradition of the Moscow mathematical school.

Yu. S. Ilyashenko

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The prominent mathematician Yulij Sergeevich Ilyashenko, the president of the Independent Moscow University, a member of the board of the Moscow Mathematical Society, and a Commodore of the Ordre des Palmes Académiques, observed his 80th birthday on 4 November 2023.

Yulij Ilyashenko’s field is dynamical systems, one of the most classical fields of mathematics, going back to Leibniz and Euler. During the last decades it transformed significantly through the work of Ilyashenko and his school.

He was born in Moscow on 4 November 1943. On graduating from Moscow school no. 59, where Ivan Vasil’evich Morozkin was his mathematics teacher, he enrolled in the Faculty of Mechanics and Mathematics at Moscow State University, from which he graduated in 1965. Evgenii Mikhailovich Landis was his scientific advisor. It was believed at that time that the pioneering work of Landis and I. G. Petrovskii (see [42]) provided a solution to the second part of Hilbert’s 16th problem. Recall its statement. Consider a system of differential equations in the plane:

The core idea of Landis and Petrovskii was to complexify the phase portrait of the vector field in question, that is, to consider complex variables $x$ and $y$ in (1) and investigate the dependence on parameters in analytic families of complex analogues of limit cycles. However, it turned out that their approach did not work in the form they proposed. This was discovered in the mid-1960s, by Ilyashenko and S. P. Novikov (who also mentioned D. V. Anosov in this connection). Nevertheless, the studies of Petrovskii and Landis marked the beginning of the theory of holomorphic foliations with singularities, a new area at the interface of analytic and algebraic geometry, complex analysis, topology, differential equations, singularity theory, and dynamical systems.

In his Ph.D. thesis “Limit cycles arising in perturbations of the equation $dw/dz=-R_{z}/R_{w}$, where $R(z,w)$ is a polynomial”, defended by Ilyashenko in 1969, he considered the problem of the bifurcation of limit cycles under a polynomial perturbation $\mathrm{d}H+\varepsilon\omega=0$ of the planar Hamiltonian system in the Pfaffian form $\mathrm{d}H=0$. It had been known since the works of Poincaré and Pontryagin that a necessary condition for a limit cycle to arise in a neighbourhood of an oval $\delta\subseteq\{H=c\}$ on a level curve of the function $H$ is the vanishing Abelian integral

In the early 1970s Ilyashenko and N. N. Nekhoroshev became co-supervisors of the seminar on ordinary differential equations organized earlier by Landis at the Faculty of Mechanics and Mathematics at Moscow University. By the end of the decade Landis departed from the management of the seminar, and Ilyashenko’s and Nekhoroshev’s seminars split. The seminar on dynamical system lead by Yulij Ilyashenko still functions these days; since the fall of 2017 it has moved to the Faculty of Mathematics at the HSE University.

The papers [15] and [16] contain an implicit question, which was subsequently called ‘Hilbert’s infinitesimal problem’: find an upper bound for the number of isolated real zeros of an Abelian integral (2). In 1984 A. G. Khovanskii and A. N. Varchenko proved a result on the uniform boundedndess of the number of such zeros, and the focus moved on finding explicit upper bounds. Ilyashenko established a few important explicit non-uniform asymptotic estimates for the number of zeros in a series of joint papers with S. Yu. Yakovenko (see [40] and the references there) and A. A. Glutsuk [10], and eventually, a uniform bound in the form of a double exponential was found by G. Binyamini, D. I. Novikov, and Yakovenko [2] in 2010.

In the 1970s Ilyashenko developed the local and global theory of analytic differential equations: he studied normal forms of singularities and the convergence of arising series, and considered the question of the algebraic solvability of local classification problems. He proved that the centre-focus problem and the Lyapunov stability problems are analytically unsolvable (see [19] and the references there). In parallel, he considered the typical properties of polynomial foliations of $\mathbb{C}P^2$ and their holonomy in a neighbourhood of the invariant line at infinity. In this connection he discovered the topological rigidity of complex foliations: in contrast to their real analogues, wich are normally topologically stable under small perturbations, you cannot deform a ‘complex phase portrait’ while preserving its topological type [20]. Ilyashenko was invited to the International Congress of Mathematicians in Helsinki (1978) to make a report on this result. About 2010 he returned to this subject and introduced the stronger notion of total rigidity. For generic quadratic vector fields (that is, systems of the form (1), where $\deg P,\deg Q\leqslant 2$) he established total rigidity in 2011, in a joint paper with V. Moldavskis [34].

One of the key components of the Petrovskii–Landis strategy was to prove that a complex limit cycle does not get destroyed, that is, it ‘can be continued arbitrarily far away’ as a function of the parameters and the base point in a transversal section. At the beginning of this century, in the pioneering work [5], joint with G. T. Buzzard and S. L. Hruska, Ilyashenko considered a discrete analogue of the statement on the preservation of a complex limit cycle, a theorem on the persistence of periodic trajectories of typical polynomial automorphisms of $\mathbb{C}^2$ under generic deformations. More precisely, in [5] the authors proved that the Kupka–Smale property is typical in the set of polynomial automorphisms of fixed degree: all periodic orbits of an automorphism are hyperbolic, and stable and unstable manifolds of any two saddle fixed points intersect transversally. They also showed there that a heteroclinic intersection of any two periodic saddle points is preserved by perturbations, that is, it can be extended to a large part of the parameter space. This was one of the first persistency results in multidimensional holomorphic dynamics.

The approach to the preservation of a complex limit cycle proposed by Ilyashenko in the late 1960s was to uniformize leaves of the foliation. The result on the uniformization of a single leaf is well known and described by the classical Poincaré–Koebe uniformization theorem. To investigate the global holomorphic foliation defined by a polynomial vector field we must know how the uniformizing mapping depends on the parameter on a transversal to leaves. In 1972 Ilyashenko showed that the union of universal covering of leaves with distinguished points lying in a transversal section admits the natural structure of a Stein manifold [17] and asked whether or not the universal covering can be realized as a domain in the Cartesian product of the transversal section and a Riemann sphere. L. Bers’s classical theorem on simultaneous uniformization states that this is possible for regular holomorphic foliations by compact Riemann surfaces such that particular leaves are uniformized by quasi-Fuchsian groups. This means that the family of universal coverings is biholomorphically equivalent to a family of invariant components of an analytic family of quasi-Fuchsian groups, so that the quotient of each component by the action of the group is a leaf of the family. In 1973 Ilyashenko proved an important result for a wider class of Kleinian groups, the so-called non-degenerate $B$-groups [18]. Subsequently, this allowed him to generalize Bers’s theorem to foliations by compact Riemann surfaces in a neighbourhood of a leaf whose singularities are simple transversal self-intersections (double points); the corresponding paper is in preparation. These results of Ilyashenko’s were an outstanding contribution to both the theory of holomorphic foliations and the classical theory of Kleinian groups. For more general foliations of algebraic surfaces counterexamples were found by Glutsyuk [9], a student of Ilyashenko, in the early 2000s.

Let us return to the 1980s. At that time Ilyashenko supervised a few postgraduate students (A. A. Scherbakov, V. A. Naishul’, P. M. Elizarov, and S. M. Voronin), who discovered surprizing phenomena in the analytic and topological singularity theory. In 1981 Voronin discovered a geometric obstruction to the convergence of a formal reduction to a normal form of a parabolic fixed point of a holomorphic automorphism [45]. This obstruction, also discovered, independently and analytically, by J. Écalle [7] and now known as the Écalle–Voronin modulus of analytic classification, has (in combination with the subsequent results of J. Martinet and J.-P. Ramis on the analytic classification of saddle nodes and resonant saddles) developed into a starting points for attack on limit cycles.

In 1980 A. Dulac’s famous memoir “Sur les cycles limites” (1923), concerned with the proof that a polynomial vector field has a finite number of finite cycles, was translated into Russian. Ilyashenko began to read it meticulously, by paying particular attention to arguments linking the formal and analytic properties of differential equations: it became clear at that time that there is a great gap between them. Very soon he came across the so-called ‘Dulac’s last lemma’, from which it followed, as he observed, that a $C^\infty$-function with asymptotic series of a certain special form (with non-oscillating terms formed by real powers and logarithms of a positive real variable) must itself be non-oscillating on a small positive interval. This is indeed true for functions with non-trivial asymptotic series (containing a leading term), but this fails for flat functions (with identically vanishing asymptotic series). Ilyashenko presented an example of a polycycle with flat monodromy of an analytic vector field on a complex manifold ([22], Theorem 4), and later a similar example was constructed in a plane domain by S. I. Trifonov [44].

Incidentally, a number of overseas experts also worried about the validity of Dulac’s proof; in 1976 F. Dumortier asked this question on the sidelines of a conference in Rio de Janeiro, and about 1981 R. Moussu sent letters to a few experts, where he wondered if they considered Dulac’s proof complete. Ilyashenko was one of these experts, and he answered by a full-scale counterexample. The first announcement of the fall of an immense structure (Dulac’s memoir is about 150 pages long) was made in a preprint of the Pushchino Scientific Center, where Ilyashenko made a report on this subject at a conference in February 1982. Then the survey paper [23] was published in 1985 in the journal Uspekhi Matematichskikh Nauk,^1[x]1Translated into English as Russian Mathematical Surveys. in which Ilyashenko reproduced almost all of Dulac’s results, but showed that these were fundamentally insufficient to prove the statement on the finite number of limit cycles. This marked the start of the search for the proof of ‘Dulac’s conjecture’.

In 1986 R. Bamon proved a finiteness theorem for limit cycles of quadratic vector fields, and in 1990 two successive reports were made at the International Congress of Mathematicians in Kyoto: one by Jean Écalle, entitled “The acceleration operators and their applications to differential equations, quasianalytic functions, and the constructive proof of Dulac’s conjecture”, and another by Yulij Ilyashenko, “Finiteness theorems for limit cycles”. Each speaker presented his own independent proof of the finiteness theorem: Écalle’s one used the techniques of acceleration of the summation process and Ilyashenko’s proof was based on the extensive use of the geometric and analytic theory of ‘cohomoogy maps’, sets of analytic functions defined on intersecting domains such that the decay of their differences on intersections is subject to explicit control. The detailed proofs were only published a few months later, as the monographs [24] and [8] of several hundred pages each. These days the Écalle–Ilyashenko finiteness theorem is fairly considered to be a peak achievement in the 100-years history of Hilbert’s 16th problem. However, the problem itself is still open: even in the simplest case of quadratic vector fields no upper bound is known for the number of limit cycles that is common to all such fields. (In 1980 S. Songling showed the existence of quadratic vector fields with at least four limit cycles).

The main source of complications in a finiteness proof is the case of a degeneracy of infinite codimension. Following V. I. Arnold’s ideas, Ilyashenko conjectured that we can avoid such degeneracies in typical finite-parameter families of vector fields, so there is an upper bound for the number of limit cycles occurring in a neighbourhood of a separatrix polygon (as follows from arguments similar to the proof of the Poincaré–Bendixson theorem, this is the only part of a phase portrait where the occurrence of an unlimited number of limit cycles is possible in principle). Under the assumption that singular points in this polygon are generic saddles or saddle nodes, this conjecture, called the Hilbert–Arnold problem, was eventually proved by Ilyashenko, in a series of joint papers with Yakovenko [39], and by V. Yu. Kaloshin [41].

In the early 1980s Ilyashenko discovered an error in J. Plemelj’s book of 1908, which was believed to contain a solution of another problem of Hilbert’s, namely, the 21st, also known as the Riemann–Hilbert problem. This problem consists in constructing a system of linear ordinary differential equations with rational coefficients that have only first-order poles at prescribed singular points $a_1,\dots,a_n$ and have a prescribed monodromy group, the group of matrices describing the isomorphisms of the solution space generated by the continuation of solutions in $\overline{\mathbb{C}}\setminus\{a_1,\dots,a_n\}$. Ilyashenko and, at about the same time and independently, A. Treibich Kohn [43] understood that Plemelj’s arguments worked only under the implicit assumption that one of these matrices is diagonalizable, so that the general case was still open. This observation prompted A. A. Bolibrukh, who worked at that time at the multidimensional version of the Riemann–Hilbert problem, to return to the one-dimensional case. In 1992 Bolibrikh discovered a very subtle topological obstruction to the solvability of the Riemann–Hilbert problem.

In the summer of 1996 L. Ortiz Bobadilla and E. Rosales González organized a seminar on dynamical systems in Cuernavaca and offered Ilyashenko to read a cycle of lectures there. Subsequently, he included the material of these lectures in the book Nonlocal bifurcations [33], written jointly with W. Li. A bifurcation of a saddle-node cycle with several homoclinic tori motivated them to consider the dynamics of semigroups of circle transformations generated by a finite number of diffeomorphisms. As a result, Ilyashenko and his students turned to stochastic dynamical systems. There they discovered some properties (for example, the stable existence of dense orbits with zero Lyapunov exponent, or the coexistence of dense sets of periodic orbits with different indices) which were quite unusual for classical dynamical systems. On the other hand, by construction these properties can be observed inside an invariant set of the original diffeomorphism.

In September of 1996, at a meeting of the Moscow Mathematical Society Ilyashenko proposed a heuristical principle: any phenomenon observed in the dynamics of a finitely generated free semigroup of diffeomorphisms can also be realized in a partially hyperbolic set of a single map. The first technical steps towards the implementation of this program were made by Ilyashenko and A. S. Gorodetski in a series of papers (see [13] and the references in that paper). Their results underlie many examples and counterexamples: open sets of diffeomorphisms possessing non-hyperbolic ergodic measures (V. A. Kleptsyn and M. B. Nalsky), skew products with non-removable zero Lyapunov exponents (Gorodetski, Ilyashenko, Kleptsyn, and Nalsky: see [14]), so-called ‘bony attractors’ (Yu. G. Kudryashov), invisible attractors (D. S. Volk, Ilyashenko, and A. Negut [38], [35]), non-density of the orbital shadowing property in the $C^1$-topology (A. V. Osipov), an open set of maps of an annulus with intermingled attraction basins (Ilyashenko, Kleptsyn, and P. S. Saltykov [30]), non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes (C. Bonatti, L. Diaz, and Gorodetski: see [6]), relatively unstable Milnor attractors (Ilyashenko and I. S. Shilin [36]), and many other results.

In the same year of 1996 the Steklov Mathematical Institute of the Academy of Sciences moved to a new building, where Ilyashenko and Anosov soon organized a seminar on dynamical systems. R. I. Grigorchuk reported there on his results on ergoodic theorems for group actions in the case of a free group, which were subsequently developed by participants of the seminar [3], [4].

Another question Ilyashenko has been dealing with throughout his life in mathematics is about the ‘correct’ identification of ‘observed modes’ of a typical dynamical system, that is, regimes to which most of its points tend as $t\to +\infty$. This limit set is called at attractor of the system, but the meaning of ‘most’ and ‘tend’ in this definition can be understood in different ways. In some cases the standard definitions for an ‘essential’ part of the phase space (for instance, the non-wandering set or the maximal attractor) include some ‘extra’ trajectories (which cannot be observed in practice in any reasonable sense). Ilyashenko introduced and investigated two definitions of an attractor, a statistical attractor and a minimal attractor ([25]; also see [13]), which are now often called the Ilyashenko statistical and minimal attractors. The statistical attractor is perhaps the best candidate for an attractor ‘observed in reality’. The minimal attractor is rigorously defined as the minimal set containing the support of any invariant measure obtained from the Lebesgue measure by the Krylov–Bogolyubov procedure, and the statistical attractor is the least closed set such that for each neighbourhood $U$ of it, for almost all points $p$ in the phase space the share of time that the trajectory $p$ remains in $U$ tends to 1 with the increase of the length of the interval of time under consideration. At about the same time J. Milnor and D. Ruelle proposed their versions of the definition of an attractor. Although there are examples showing that these definitions can produce different sets, the question of the relationship between different attractors of a typical dynamical system is still open. Subsequently, this line of research was developed further by Ilyashenko with his students P. S. Bachurin, A. V. Okunev, N. A. Solodovnikov, and I. S. Shilin, and, in particular, examples of intermingled attraction basins and so-called ‘thick attractors’ (see [26]) were produced.

Ilyashenko found also an upper bound for the Hausdorff and Minkowski dimensions of the attractors of dissipative dynamical systems used in investigations of the Navier–Stokes and Kuramoto–Sivashinski equations [21], [29], [25], [1].

Now we describe Ilyashenko’s results of recent years, heralding the opening of a new chapter in the theory of bifurcations of vector fields on 2-surfaces. It is known that typical vector fields on $S^2$ are structurally stable. However, a similar question on finite-parameter families of vector fields is much less trivial. In 1981 I. P. Malta and J. Palis observed that one-parameter bifurcations of a planar vector field with semistable cycle of multiplicity two and several separatrices winding about this cycle have numerical classification invariants. Their proof was based on the fact that for such vector fields the Poincaré maps on a transversal to the cycle form a saddle-node family of diffeomorphisms of an interval, and families of this type have a rather rigid structure. Subsequently, R. Roussarie proved on this basis that three-parameter bifurcations of a planar vector field with semistable cycle of multiplicity four have functional invariants. All these cases are related to the classification of families up to strong equivalence: two families of vector fields $(u_\delta)$ and $(v_\varepsilon)$ are strongly equivalent if, after an appropriate substitution $\delta=h(\varepsilon)$ in the parameter space, the flows of the fields $u_{h(\varepsilon)}$ and $v_\varepsilon$ are conjugate by means of a homeomorphism $H_\varepsilon$, and this homeomorphism depends continuously on the parameter $\varepsilon$.

On the basis of these examples, in the early 1990s Arnold proposed a much weaker equivalence relation, which, as he conjectured, could be insensitive to such effects. He conjectured that the classification of typical finite-parameter families of vector fields on a sphere in terms of this weak equivalence if very simple: typical families are structurally stable, and each vector field in such a family has a finite-parameter versal deformation that contains information on all possible perturbations of this field. Less formally, a bifurcation in a typical family depends on the most degenerate field in this family, and there is only a finite number of types of such fields for a given dimension of the parameter space.

In 2018 Ilyashenko with his students Kudryashov and I. V. Schurov [32] refuted Arnold’s conjecture for three-parameter families. They discovered an unexpected phenomenon in the case of a vector field with two saddle points $L$ and $M$ whose separatrices close up to form a heart-like shape (the two common separatrices of $L$ and $M$) and a ‘water drop’ (another separatrix going out from $M$ and coming in back to $M$). Such a polycycle (called affectively ‘a heart with a tear’) occurs in three-parameter families in an unremovable way, and if one adds here a pair of saddle points, outside the polycycle and inside the ‘tear’, then the bifurcation has a numerical invariant, the ratio of the characteristic numbers of these saddles. In addition, the authors showed that the space of six-parameter families contains an open subset of families with functional invariants. These breakthrough results were based on an ingenious observation of Ilyashenko: when a separatrix loop (in this case the one forming the ‘tear’) opens up, two sequences $(e_n)$ and $(i_n)$ of values of the parameter arise, both tending to zero, for which one of the separatrices winding about the cycle closes up with one of the separatrices of $M$ that previousy formed a loop: such alternating closures of separatrices are called sparkling separatrix connections. Now a numerical invariant reflects the mutual position of the sequences $(e_n)$ and $(i_n)$, namely, their relative density.

In the same paper [32] they introduced a new equivalence relation between families of vector fields, moderate topological equivalence, which is intermediate between strong and weak equivalences and is apparently the most adequate expression of the intuitive notion of ‘two families with the same bifurcations’. The authors proved the existence of whole domains of families of vector fields that are structurally unstable in the sense of moderate equivalence, and they conjectured that these families are also inequivalent in the sense of weak equivalence. Structural instability is here due to the fact that separatrices of saddles not included in the polycycle wind about it, so that the bifurcation depends on the behaviour of the unperturbed vector field ouside a neighbourhood of the polycycle. Ilyashenko called this new type of bifurcation glocal bifurcations, to distinguish them from local bifurcations occurring in a small neighbourhood of a degenerate point and from semilocal ones which occur in a neighbouhood of a limit cycle or a semicycle: there we only have ‘locality’ with respect to a parameter.

As we see from the above discussion, global bifurcations have a more complicated structure than Arnold believed, and cannot be classified very simply. In [27] and [32] Ilyashenko put forward a program of further research: develop a theory of global bifurcations of vector fields on a 2-sphere. Now we describe some results obtained in the framework of this program.

In an analysis of a bifurcation one of the decisive steps is usually finding its support, a set such that the behaviour of vector fields in a neighbourhood of it enables us to describe fully the bifurcation in the family. One would like to have a support as small as possible. Remarkably, Ilyashenko and N. B. Goncharuk [11] managed to describe explicitly such a set, which they called a large bifurcation support (‘large’ is due to the fact that this support is larger than Arnold supposed), and moreover, for an arbitrary finite-parameter family of vector fields on a 2-sphere. By considering the large support we can localize the set undergoing the bifurcation and compare the global behaviour of different families of vector fields.

Ilyashenko and his students have advanced even further since then. For example, they proved that typical one-parameter families are structurally stable (V. V. Starichkova, Goncharuk, Ilyashenko, and N. A. Solodovnikov [37], [12]) and found an example of a five-parameter family with functional invariants (Goncharuk and Kudryashov).

Many questions in Ilyashenko’s program still remain open. The bifurcation diagrams for three-parameter families, which were described in [32], are poorly understood. In many new examples the full lists of invariants are not yet known. The description of all typical two-parameter families and the verification of their structural stability are far from completion. However, there seems to be some progress in the last issue: as Ilyashenko showed in his recent paper [28], a certain rather complex family of two-parameter bifurcations on $S^2$ is structurally stable.

In different periods of time Ilyashenko was a professor of Moscow and Cornell Universities, and a researcher in the Steklov Mathematical Institute. Currently he is the president of the Independent University of Moscow and a research professor in the Faculty of Mathematics at the HSE University, of which he is one of the founders. He is a member of the editorial boards of the journals Trydy Moskovskogo Matematicheskogo Obshchestva,^2[x]2Translated in to English as Proceeding of the Moscoew Mathematical Society. Matematicheskoe Prosveshchenie, and Journal of Dynamical and Control Systems. For many years he was a member of the editorial board of the journal Funktsional’nyi Analiz i ego Prilozheniya.^3[x]3Translated into English as Functional Analysis and its Applications.

In 2001 Ilyashenko and M. A. Tsfasman organized the ‘Globus’ seminar in the Independent University of Moscow, and in the same year they founded the Moscow Mathematical Journal, of which they (together with S. M. Gusein-Zade) still are the editors-in-chief. For many years Ilyashenko has been a member of the jury of contests for young mathematicians: the August Möbius Contest, and the contests of Pierre Deligne, the “Dynasty” Foundation, and the “Junior Mathematics of Russia”.

For his research achievements and the key role he played in the development of cooperation between Russian and French mathematicians Yulij Ilyashenko received state awards of the French Republic: he is a Chevalier (2005) and a Commodore (2018) of the Ordre des Palmes Académiques.

Ilyashenko is an exceptional teacher, whose lectures are very popular because of their clarity and rigour. Detailed, meticulously elaborated educational materials of Ilyashenko testify to his wish to present complicated subjects to students in the most accessible way. For many years he read a course of ordinary differential equations in the Faculty of Mechanics and Mathematics at Moscow State University, and there are plans to publish a textbook on this basis.^4[x]4A. I. Bufetov, N. B. Goncharuk, and Yu. S. Ilyashenko, Ordinary differential equations. The current version (in Russian) is accessible on https://math.hse.ru/difury_ilyashenko2023. Starting in 1993, Ilyashenko has participated in the work of the organizing committee of the Moscow Mathematical Olympiad. In 1993 special lectures were organized for the winners of the olympiad, and Ilyashenko read there a lecture on the Smale horseshoe, published subsequently in an article (joint with A. Yu. Kotova) in the journal Kvant [31]. To the best of our knowledge, this was the first story about the Smale horseshoe aimed at high-school students. Some listeners to that lecture have since then developed into experts in dynamical systems. Ilyashenko reads lectures at the summer schools “Contemporary Mathematics” in Ratmino; he wrote two pamphlets Attractors and their fractal dimension and Evolutionary processes and the philosophy of general position on the basis of these lectures.

Ilyashenko was the scientific advisor of the Ph.D. theses of O. D. Anosova, A. M. Arkhipov, A. R. Borisyuk, D. S. Volk, S. M. Voronin, N. Dimitrov, A. V. Dukov, P. M. Elizarov, A. A. Glutsyuk, T. I. Golenishcheva-Kutuzova, N. B. Goncharuk, I. Gorbovickis, A. S. Gorodetski, J. A. Jaurez Rosas, P. I. Kaleda, V. A. Kleptsyn,^5[x]5Joint supervision with É. Ghys. G. A. Kolyutskii,^6[x]6Joint supervision with D. V. Anosov. Yu. G. Kudryashov,^5[x]5Joint supervision with É. Ghys. N. B. Medvedeva, S. S. Minkov, V. Moldavskis, B. Müller, M. B. Nalsky, A. V. Okunev, L. Ortiz Bobadilla, A. A. Panov, I. A. Pushkar’, V. Ramírez, O. L. Romaskevich,^5[x]5Joint supervision with É. Ghys. P. S. Saltykov, N. A. Solodovnikov, V. V. Stantso, S. I. Trifonov, D. A. Filimonov, A. Yu. Fishkin, I. S. Shilin, I. V. Schurov, A. A. Shcherbakov, and S. Yu. Yakovenko; among the students who prepared their Master theses under Ilyashenko’s supervison were I. A. Androsov, T. N. Bakiev, P. S. Bachurin, A. I. Bufetov, A. A. Dorovskii, V. Yu. Kaloshin, A. Yu. Kotova, M. E. Saprykina, V. V. Starichkova, R. M. Fëdorov, and B. L. Shleifman. There are many mathematicians who regard Yulij Ilyashenko as their Teacher.

In the 1990s, when many of his colleagues departed from Russia, Ilyashenko was one of the world-class researchers who remained accessible to students of Moscow University as a potential scientific advisor. He believes it to be his obligation to reveal the beauty of the whole mathematics to his students and give them a possibly broader mathematical education. Together with his wife Elena Nikolaevna, who supports Yulij Ilyashenko in all of his undertakings, he helps his students by his wise encouragement at crucial moments of their lives. He is a moral compass for his students.

Ilyashenko’s famous Friday seminar on dynamical systems confidently enters its sixth decade. Following a proposal of Gorodenski, since 1998 Ilyashenko has been organizing annual summer schools for its participants; in recent years the location was in Ratmino, on the premises of the Joint Institute for Nuclear Energy. Mornings begin with a two-hour lecture, or a pair of lectures read by Ilyashenko. Then other participants, including students, report on their recent results. For the junior participants basic courses are organized, where they are dealt out exercises in the spirit of schoolchildren’ study groups, and then juniors submit their assignments to seniors. In this way newcomers can rapidly immerse into the topics discussed at the seminar.

Among Ilyashenko’s broad interests outside mathematics we can distinguish history and poetry. The seminar organizes poetry evenings and role readings. The participants remember it well how Ilyashenko himself read M. Lermontov’s and A. Akhmatova’s poems and how people from the seminar, together with Ilyashenko’s grandson Fëdor Rodin, read scenes from Pushkin’s Boris Godunov. At summer schools participants attend evening ‘humanities lectures’ (as Ilyashenko calls them) on most diverse subjects, from Donatello to the history of cartoons, from Plato to details of the tea ceremony.

Yulij Ilyashenko is a genuine inspirational leader of his large scientific family. Each of us has been strongly influenced by his bright personality. By his research, teaching, and multifaceted organizational activities he makes his unique contribution to the development of mathematics in Russia. We congratulate Yulij Sergeevich cordially on his eightieth birthday and wish him new lecture courses, new schools and conferences, new theorems, new students, and new books!

Bibliography
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2.	G. Binyamini, D. Novikov, and S. Yakovenko, “On the number of zeros of Abelian integrals”, Invent. Math., 181:2 (2010), 227–289
3.	A. I. Bufetov, “Convergence of spherical averages for actions of free groups”, Ann. of Math. (2), 155:3 (2002), 929–944
4.	A. I. Bufetov, A. Klimenko, and C. Series, “Convergence of spherical averages for actions of Fuchsian groups”, Comment. Math. Helv., 98:1 (2023), 41–134
5.	G. T. Buzzard, S. L. Hruska, and Yu. Ilyashenko, “Kupka–Smale theorem for polynomial automorphisms of $\mathbb C^2$ and persistence of heteroclinic intersections”, Invent. Math., 161:1 (2005), 45–89
6.	L. J. Díaz and A. Gorodetski, “Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes”, Ergodic Theory Dynam. Systems, 29:5 (2009), 1479–1513
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8.	J. Écalle, Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac, Actualités Math., Hermann, Paris, 1992, ii+340 pp.
9.	A. Glutsyuk, “Nonuniformizable skew cylinders. A counterexample to the simultaneous uniformization problem”, C. R. Acad. Sci. Paris Sér. I Math., 332:3 (2001), 209–214
10.	A. A. Glutsyuk and Yu. S. Ilyashenko, “Restricted version of the infinitesimal Hilbert 16th problem”, Mosc. Math. J., 7:2 (2007), 281–325
11.	N. Goncharuk and Yu. Ilyashenko, Large bifurcation supports, 2019 (v1 – 2018), 74 pp., arXiv: 1804.04596
12.	N. Goncharuk, Yu. Ilyashenko, and N. Solodovnikov, “Global bifurcations in generic one-parameter families with a parabolic cycle on $S^2$”, Mosc. Math. J., 19:4 (2019), 709–737
13.	A. S. Gorodetski and Yu. S. Ilyashenko, “Certain properties of skew products over a horseshoe and a solenoid”, Proc. Steklov Inst. Math., 231 (2000), 90–112
14.	A. S. Gorodetski, Yu. S. Ilyashenko, V. A. Kleptsyn, and M. B. Nalsky, “Nonremovable zero Lyapunov exponents”, Funct. Anal. Appl., 39:1 (2005), 21–30
15.	Ju. S. Il'jas̆enko (Ilyashenko), “The origin of limit cycles under perturbation of the equation $dw/ dz=-R_z/ R_w$, where $R(z,w)$ is a polynomial”, Math. USSR-Sb., 7:3 (1969), 353–364
16.	Ju. S. Il'jas̆enko (Ilyashenko), “An example of equations $dw/dz=P_n(z,w)/Q_n(z,w)$ having a countable number of limit cycles and arbitrarily large Petrovskiĭ–Landis genus”, Math. USSR-Sb., 9:3 (1969), 365–378
17.	Ju. S. Il'jas̆enko, “Fiberings into analytic curves”, Math. USSR-Sb., 17:4 (1972), 551–569
18.	Ju. S. Il'jas̆enko (Ilyashenko), “Nondegenerate $B$-groups”, Soviet Math. Dokl., 14 (1973), 207–210
19.	Ju. S. Il'jas̆enko (Ilyashenko), “Analytic unsolvability of the stability problem and the problem of topological classification of the singular points of analytic systems of differential equations”, Math. USSR-Sb., 28:2 (1976), 140–152
20.	Yu. Il'yashenko, “The topology of phase portraits of analytic differential equations in the complex projective plane”, Selecta Math. Soviet., 5:2 (1986), 141–199
21.	Yu. S. Il'yashenko, “Weakly contracting systems and attractors of Galerkin approximations of Navier–Stokes equations on the two-dimensional torus”, Selecta Math. Soviet., 11:3 (1992), 203–239
22.	Yu. S. Il'yashenko, “Limit cycles of polynomial vector fields with nondegenerate singular points on the real plane”, Funct. Anal. Appl., 18:3 (1984), 199–209
23.	Yu. S. Ilyashenko, “Dulac's memoir “On limit cycles” and related problems of the local theory of differential equations”, Russian Math. Surveys, 40:6 (1985), 1–49
24.	Yu. S. Il'yashenko, Finiteness theorems for limit cycles, Transl. from the Russian, Transl. Math. Monogr., 94, Amer. Math. Soc., Providence, RI, 1991, x+288 pp.
25.	Ju. S. Il'yashenko, “The concept of minimal attractor and maximal attractors of partial differential equations of the Kuramoto–Sivashinsky type”, Chaos, 1:2 (1991), 168–173
26.	Yu. Ilyashenko, “Thick attractors of boundary preserving diffeomorphisms”, Indag. Math. (N. S.), 22:3-4 (2011), 257–314
27.	Y. Ilyashenko, “Towards the general theory of global planar bifurcations”, Mathematical sciences with multidisciplinary applications., In honor of professor Christiane Rousseau, and in recognition of the Mathematics for Planet Earth initiative, ed. B. Toni, Springer Verlag, Cham, 2016, 269–299
28.	Yu. Ilyashenko, “Germs of bifurcation diagrams and SN–SN families”, Chaos, 31:1 (2021), 013103, 15 pp.
29.	Iu. S. Il'iashenko and A. N. Chetaev, “On the dimension of attractors for a class of dissipative systems”, J. Appl. Math. Mech., 46:3 (1982), 290–295
30.	Yu. S. Ilyashenko, V. A. Kleptsyn, and P. Saltykov, “Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins”, J. Fixed Point Theory Appl., 3:2 (2008), 449–463
31.	Yu. Ilashenko and A. Kotova, “Smale horseshoe”, Kvant, 1994, no. 1, 15–19 (Russian)
32.	Yu. Ilyashenko, Yu. Kudryashov, and I. Schurov, “Global bifurcations in the two-sphere: a new perspective”, Invent. Math., 213:2 (2018), 461–506
33.	Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, Math. Surveys Monogr., 66, Amer. Math. Soc., Providence, RI, 1999, xiv+286 pp.
34.	Yu. Ilyashenko and V. Moldavskis, “Total rigidity of generic quadratic vector fields”, Mosc. Math. J., 11:3 (2011), 521–530
35.	Yu. Ilyashenko and A. Negut, “Invisible parts of attractors”, Nonlinearity, 23:5 (2010), 1199–1219
36.	Yu. S. Ilyashenko and I. S. Shilin, “Relatively unstable attractors”, Proc. Steklov Inst. Math., 277 (2012), 84–93
37.	Yu. Ilyashenko and N. Solodovnikov, “Global bifurcations in generic one-parameter families with a separatrix loop on $S^2$”, Mosc. Math. J., 18:1 (2018), 93–115
38.	Yu. Ilyashenko and D. Volk, “Cascades of $\varepsilon$-invisibility”, J. Fixed Point Theory Appl., 7:1 (2010), 161–188
39.	Yu. S. Il'yashenko and S. Y. Yakovenko (eds.), Concerning the Hilbert 16th problem, Amer. Math. Soc. Transl. Ser. 2, 165, Adv. Math. Sci., 23, Amer. Math. Soc., Providence, RI, 1995, viii+219 pp.
40.	Yu. Il'yashenko and S. Yakovenko, “Double exponential estimate for the number of zeros of complete Abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group”, Invent. Math., 121:3 (1995), 613–650
41.	V. Kaloshin, “The existential Hilbert 16-th problem and an estimate for cyclicity of elementary polycycles”, Invent. Math., 151:3 (2003), 451–512
42.	E. M. Landis and I. G. Petrovskiĭ, “On the number of limit cycles of the equation $\frac{dy}{dx}=\frac{P(x,y)}{Q(x,y)}$, where $P$ and $Q$ are polynomials”, Amer. Math. Soc. Transl. Ser. 2, 14, Amer. Math. Soc., Providence, RI, 1960, 181–199
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Citation in format AMSBIB

\Bibitem{BakBufVas25}
\by T.~Bakiev, A.~I.~Bufetov, V.~A.~Vassiliev, S.~M.~Voronin, A.~A.~Glutsyuk, A.~S.~Gorodetski, A.~V.~Dukov, V.~Yu.~Kaloshin, A.~V.~Klimenko, V.~V.~Kozlov, S.~B.~Kuksin, S.~K.~Lando, V.~S.~Oganesyan, G.~I.~Olshanski, S.~Yu.~Pilyugin, O.~V.~Pochinka, Ya.~G.~Sinai, A.~S.~Skripchenko, A.~L.~Skubachevskii, I.~A.~Taimanov, V.~A.~Timorin, V.~M.~Tikhomirov, D.~V.~Treschev, D.~A.~Filimonov, K.~M.~Khanin, H.~Hedenmalm, A.~G.~Khovanskii, M.~A.~Tsfasman, A.~I.~Shafarevich, I.~S.~Shilin, S.~Yu.~Yakovenko
\paper On the 80th birthday of Yulij Sergeevich Ilyashenko
\jour Russian Math. Surveys
\yr 2025
\vol 80
\issue 2
\pages 345--357
\mathnet{http://mi.mathnet.ru/eng/rm10227}
\crossref{https://doi.org/10.4213/rm10227e}
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\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2025RuMaS..80..345B}
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