Yu. S. Ilyashenko

Centennial History of Hilbert’s 16th Problem

The second part of Hilbert’s 16th problem deals with polynomial differential equations in the plane. It remains unsolved even for quadratic polynomials. There were several attempts to solve it that failed. Yet the problem inspired significant progress in the geometric theory of planar differential equations, as well as bifurcation theory, normal forms, foliations and some topics in algebraic geometry. The dramatic history of the problem, as well as related developments, are presented below. §1. The problem and its counterparts What may be said about the number and location of limit cycles of a planar polynomial vector field of degree n? (The limit cycle is an isolated closed orbit of a vector field.) This second part of Hilbert’s 16th problem appears to be one of the most persistent in the famous Hilbert list [H], second only to the Riemann ζ-function conjecture. Traditionally, Hilbert’s question is split into three, each one requiring a stronger answer. Problem 1. Is it true that a planar polynomial vector field has but a finite number of limit cycles? Problem 2. Is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only? The bound on the number of limit cycles in Problem 2 is denoted by H(n) and known as the Hilbert number. Linear vector fields have no limit cycles; hence H(1) = 0. It is still unknown whether or not H(2) exists. Problem 3. Give an upper bound for H(n). A solution to any of these problems implies a solution for the previous ones. Only the first problem is solved now. The positive answer was established in [E92], [I91]. There are analytic counterparts of Problems 1 and 2. Received by the editors December 2001. 2000 Mathematics Subject Classification. Primary 34Cxx, 34Mxx, 37F75.

Hilbert-type numbers for Abel equations, growth and zeros of holomorphic functions

In this paper an upper estimate of the number of limit cycles of the Abel equation = v(x,t), x , t S1 is given. Here v is a polynomial in x with the higher coefficient one and periodic in t with period one. The bound depends on the degree n of the polynomial and the magnitude of its coefficients. In the second part we give an explicit upper estimate of the number of zeros of a holomorphic function in a compact subset of its domain through the growth rate of the function and some geometric constant that is expressed here by means of the Poincare metric. This improves the estimate given in Ilyashenko and Yakovenko (1996 J. Differ. Equ. 126 87-105).

Certain new robust properties of invariant sets and attractors of dynamical systems

UDC 517.938 w Introduction In this paper, new robust properties of (partially hyperbolic) invariant sets and attractors of diffeomorphisms are obtained, including the coexistence of dense sets of periodic points with different indices and the existence of a dense orbit with zero Lyapunov exponent. Tlm precise statements of these properties are given below in Theorem A. Partially hyperbolic invariant sets with the property of robust transitivity were intensively studied [3-7]. A review of this field with a vast bibliography can be found in [8].

MINIMAL AND STRANGE ATTRACTORS

A general concept going back to Kolmogorov claims that if a dynamical system has a complicated attracting set then its behavior has not a deterministic, but rather probabilistic character. This concept was not formalized up to now. Even the definition of attractor has a lot of different versions. This paper presents an attempt to give some definitions and results formalizing this heuristic ideas. It contains a definition of a minimal attractor, modifying the one given in Ilyashenko [1991]. The actual minimality of the attractor is discussed. The principal result is the Triple Choice Theorem. It claims that the existence of a strange minimal attractor implies some mild form of chaos for the map itself or for a nearby one. The program of further investigation is proposed as a chain of problems at the end of the paper.

Some open problems in real and complex dynamical systems

Theory of dynamical systems may be split into two parts. The larger one, dealing with multidimensional systems: flows in dim 3 and higher, diffeomorphisms in dim 2 and higher, may be called the realm of chaos. The smaller one, dealing with planar differential equations, may be called the realm of order. The problems below deal with both parts.

Thick attractors of step skew products

A diffeomorphism is said to have a thick attractor provided that its Milnor attractor has positive but not full Lebesgue measure. We prove that there exists an open set in the space of boundary preserving step skew products with a fiber [0,1], such that any map in this set has a thick attractor.

Persistence theorems and simultaneous uniformization

One of the most intriguing problems in the theory of foliations by analytic curves is that of the persistence of complex limit cycles of a polynomial vector field, as well as related problems concerning the persistence of identity cycles and saddle connections and the global extendability of the Poincaré map. It is proved that all these persistence problems have positive solutions for any foliation admitting a simultaneous uniformization of leaves. The latter means that there exists a uniformization of leaves that analytically depends on the initial condition and satisfies certain additional assumptions, called continuity and boundedness. Thus, the results obtained are conditional, but they establish a relation between very different properties of foliations.

Finiteness theorems for limit cycles: A digest of the revised proof

This is the first paper in a series of two presenting a digest of the proof of the finiteness theorem for limit cycles of a planar polynomial vector field. At the same time we sketch the proof of the following two theorems: an analogous result for analytic vector fields, and a description of the asymptotics of the monodromy transformation for polycycles of such fields.

Relatively unstable attractors

There are different non-equivalent definitions of attractors in the theory of dynamical systems. The most common are two definitions: the maximal attractor and the Milnor attractor. The maximal attractor is by definition Lyapunov stable, but it is often in some ways excessive. The definition of Milnor attractor is more realistic from the physical point of view. The Milnor attractor can be Lyapunov unstable though. One of the central problems in the theory of dynamical systems is the question of how typical such a phenomenon is. This article is motivated by this question and contains new examples of so-called relatively unstable Milnor attractors. Recently I. Shilin has proved that these attractors are Lyapunov stable in the case of one-dimensional fiber under some additional assumptions. However, the question of their stability in the case of multidimensional fiber is still an open problem.

Total rigidity of polynomial foliations on the complex projective plane

Polynomial foliations of the complex plane are topologically rigid. Roughly speaking, this means that the topological equivalence of two foliations implies their affine equivalence. There exist various nonequivalent formalizations of the notion of topological rigidity. Generic polynomial foliations of fixed degree have the so-called property of absolute rigidity, which is the weakest form of topological rigidity. This property was discovered by the author more than 30 years ago. The genericity conditions imposed at that time were very restrictive. Since then, this topic has been studied by Shcherbakov, Gómez-Mont, Nakai, Lins Neto-Sad-Scárdua, Loray-Rebelo, and others. They relaxed the genericity conditions and increased the dimension. The main conjecture in this field states that a generic polynomial foliation of the complex plane is topologically equivalent to only finitely many foliations. The main result of this paper is weaker than this conjecture but also makes it possible to compare topological types of distant foliations.