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Yulij Ilyashenko

Publications and source records attributed to Yulij Ilyashenko.

12 recordsLinked to original sources
arXiv2026

Diophantine "Tears of the Heart"

Recent studies of topologically generic unfoldings of vector fields featuring a "tears of the heart" polycycle with one internal and one external winding separatrix have shown that, in a special one-parameter subfamily where the "heart" is preserved and the "tear" loop if broken, at least four invariants of weak topological classification appear. In this paper, we demonstrate that the metrical perspective yields a different result: for Lebesgue almost all values of the coefficients related to the original vector field, the special one-parameter family generates only two such invariants.

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arXiv2025

New Numerical Invariants of an Unfolding of a Polycycle "Tears of the Heart"

In this paper new numerical invariants of structurally unstable vector fields in the plane are found. One of the main tools is an improved asymptotics of sparkling saddle connections that occur when a separatrix loop of a hyperbolic saddle breaks. Another main tool is a new topological invariant of two arithmetic progressions, both perturbed and unperturbed, on the real line. For the pairs of the unperturbed arithmetic progressions we give a complete topological classification.

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arXiv2019

Large bifurcation supports

In the study of global bifurcations of vector fields on $S^2$, it is important to distinguish a set "where the bifurcation actually occurs", -- the bifurcation support. Hopefully, it is sufficient to study the bifurcation in a neighborhood of the support only. The first definition of bifurcation support was proposed by V.Arnold. However this set appears to be too small. In particular, the newly discovered effect, an open domain in the space of three-parametric families on $S^2$ with no structurally stable families, is not visible in a neighborhood of the bifurcation support. In this article, we give a new definition of "large bifurcation support" that accomplishes the task. Roughly speaking, if we know the topological type of the phase portrait of a vector field, and we also know the bifurcation in a neighborhood of the large bifurcation support, then we know the bifurcation on the whole sphere.

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arXiv2016

Global bifurcations in the two-sphere: a new perspective

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification defined below has a numerical invariant that may take an arbitrary positive value. Here and below "families" are "families of vector fields in the two-sphere". This result disproves an Arnold's conjecture of 1985. Then we construct an open set of six parameter families whose moderate topological classification has a functional invariant. This invariant is an arbitrary germ of a smooth map $(\mathbb R_+, a)\to(\mathbb R_+, b)$. More generally, for any positive integers $d$ and $d'$, we construct an open set of families whose topological classification has a germ of a smooth map $\left(\mathbb R_+^d, a\right)\to\left(\mathbb R_+^{d'}, b\right)$ as an invariant. Any smooth germ of this kind may be realized as such an invariant. These results open a new perspective of the global bifurcation theory in the two sphere. This perspective is discussed at the end of the paper.

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arXiv2015

Sternberg linearization theorem for skew products

We present a new kind of normalization theorem: linearization theorem for skew products. The normal form is a skew product again, with the fiber maps linear. It appears, that even in the smooth case, the conjugacy is only Hölder continuous with respect to the base. The normalization theorem mentioned above may be applied to perturbations of skew products and to the study of new persistent properties of attractors.

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arXiv2010

Total rigidity of generic quadratic vector fields

We consider a class of foliations on the complex projective plane that are determined by a quadratic vector field in a fixed affine neighborhood. Such foliations, as a rule, have an invariant line at infinity. Two foliations with singularities on $\mathbb C P^2$ are topologically equivalent provided that there exists a homeomorphism of the projective plane onto itself that preserves orientation both on the leaves and in $\mathbb C P^2$ and brings the leaves of the first foliation to that of the second one. We prove that a generic foliation of this class may be topologically equivalent to but a finite number of foliations of the same class, modulo affine equivalence. This property is called \emph{total rigidity}. Recent result of Lins Neto implies that the finite number above does not exceed 240. This is the first of the two closely related papers. It deals with the rigidity properties of quadratic foliations, whilst the second one studies the foliations of higher degree.

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arXiv2010

Cascades of e-invisibility

We consider statistical attractors of locally typical dynamical systems and their "e-invisible" subsets: parts of the attractors whose neighborhoods are visited by orbits with an average frequency of less than e << 1. For extraordinarily small values of e (say, smaller than 2^(-10^6)), an observer virtually never sees these parts when following a generic orbit. A trivial reason for e-invisibility in a generic dynamical system may be either a high Lipshitz constant (~1/e) of the mapping (i.e. it badly distorts the metric) or its proximity (~e) to the structurally unstable dynamical systems. However Ilyashenko and Negut [IN] provided a locally typical example of dynamical systems with an e-invisible set and a uniform moderate (<100) Lipshitz constant independent on e. These dynamical systems from [IN] are also |log e|^{-1}-distant from structurally unstable dynamical systems (in the class S of skew products). We further develop the example of [IN] to provide a better rate of invisibility while staying at the same distance away from the structurally unstable dynamical systems. We give an explicit example of C^1-balls in the space of "step" skew products over the Bernoulli shift such that for each dynamical system from this ball a large portion of the statistical attractor is invisible. Systems that are c/n-distant from structurally unstable ones (in the class S) have rate of invisibility e = 2^(-n^k) where 3k is the Hausdorff dimension of the phase space.

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arXiv2010

Holder properties of perturbed skew products and Fubini regained

In 2006, A. Gorodetski proved that central fibers of perturbed skew products are Holder continuous with respect to the base point. In the present paper we give an explicit estimate of the Holder exponent mentioned above. Moreover, we extend the Gorodetski theorem from the case when the fiber maps are close to the identity to a much wider class that satisfy the so-called modified dominated splitting condition. In many cases (for example, in the case of skew products over the solenoid or over linear Anosov diffeomorphisms of a torus), the Holder exponent is close to 1. This allows us in a sense to overcome the so-called Fubini nightmare. Namely, we prove that the union of central fibers that are strongly atypical from the point of view of the ergodic theory, has Lebesgue measure zero, despite the lack of absolute continuity of the holonomy map for the central foliation. For that we revisit the Hirsch-Pugh-Shub theory, and estimate the contraction constant of the graph transform map.

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arXiv2009

A restricted version of the Hilbert's 16th problem for quadratic vector fields

The restricted version of the Hilbert 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields $"σ$--distant from centers and $\ka $-distant from singular quadratic vector fields" provided that the limit cycles are $"δ$--distant from singular points and infinity".

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arXiv2009

Invisible Parts of Attractors

This paper deals with the attractors of generic dynamical systems. We introduce the notion of epsilon-invisible set, which is an open set in which almost all orbits spend on average a fraction of time no greater than epsilon. For extraordinarily small values of epsilon (say, smaller than 2^{-100}), these are areas of the phase space which an observer virtually never sees when following a generic orbit. We construct an open set in the space of all dynamical systems which have an epsilon-invisible set that includes parts of attractors of size comparable to the entire attractor of the system, for extraordinarily small values of epsilon. The open set consists of C^1 perturbations of a particular skew product over the Smale-Williams solenoid. Thus for all such perturbations, a sizable portion of the attractor is almost never visited by generic orbits and practically never seen by the observer.

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arXiv2006

Variation of argument and Bernstein index for holomorphic functions on Riemann surfaces

An upper bound of the variation of argument of a holomorphic function along a curve on a Riemann surface is given. This bound is expressed through the Bernstein index of the function multiplied by a geometric constant. The Bernstein index characterizes growth of the function from a smaller domain to a larger one. The geometric constant in the estimate is explicitly given. This result is applied in \cite {GI} to the solution of the restricted version of the infinitesimal Hilbert 16th problem, namely, to upper estimates of the number of zeros of abelian integrals in complex domains.

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