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We classify global bifurcations in generic one-parameter local families of \vfs on $S^2$ with a parabolic cycle. The classification is quite different from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable.

The author was partially supported by the grants NSF 0700973 and CNRS-RFBR 10-01-93115-NTSNILa. Complex limit cycle located in a neighborhood of a hyperbolic polycycle can not vanish under a small deformation that preserves the characteristic values of the vertexes of the polycycle. The cycles either change holomorphically under the change of the parameter, or come to the boundary of the fixed neighborhood of the polycycle. The present paper makes these statements rigorous and proves them.

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.

In this paper we study attractors of skew products, for which the following dichotomy is ascertained. These attractors either are not asymptotically stable or possess the following two surprising properties. The intersection of the attractor with some invariant submanifold does not coincide with the attractor of the restriction of the skew product to this submanifold but contains this restriction as a proper subset. Moreover, this intersection is thick on the submanifold, that is, both the intersection and its complement have positive relative measure. Such an intersection is called ab one , and the attractor itself is said to be bony. These attractors are studied in the space of skew products. They have the important property that, on some open subset of the space of skew products, the set of maps with such attractors is, in a certain sense, prevalent, i.e., "big." It seems plausible that attractors with such properties also form a prevalent subset in an open subset of the space of diffeomorphisms.

In 2006, Gorodetski proved that central fibres of perturbed skew products are Holder continuous with respect to the base point. In this paper, we give an explicit estimate of this Holder exponent. Moreover, we extend Gorodetski's result from the case when the fibre maps are close to the identity to a much wider class of maps 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 the torus), the Holder exponent is close to 1. This allows one to overcome the so-called Fubini nightmare, in some sense. Namely, we prove that the union of central fibres that are strongly atypical from the point of view of ergodic theory, has Lebesgue measure zero despite the lack of absolute continuity of the holonomy map for the central foliation. This result is based on a new kind of ergodic theorem, which we call special. To prove our main result, we revisit the theory of Hirsch, Pugh and Shub, and estimate the contraction constant of the graph transform map.

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.

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.

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., 29 pages, 3 figures

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 $"\sigma $--distant from centers and $\ka $-distant from singular quadratic vector fields" provided that the limit cycles are $"\delta $--distant from singular points and infinity".

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.

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 H\"older 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., Comment: 17 pages

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, Gomez-Mont, Nakai, Lins Neto-Sad-Scardua, 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.

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 Poincare 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.

Skew products over a Bernoulli shift with a circle fiber are studied. We prove that in the space of such products there exists a nonempty open set of mappings each of which possesses an invariant ergodic measure with one of the Lyapunov exponents equal to zero. The conjecture that the space of C 2 -diffeomorphisms of the 3-dimensional torus into itself has a similar property is discussed.

Three classical results of A.A. Bolibrukh in the theory of linear systems with complex time are presented: the negative solution of the 21st Hilbert problem, sufficient conditions for this problem to have a positive solution, and sufficient conditions for the reducibility of a system with an irregular singular point to Birkhoff standard form.

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 $$(\mathbb {R}_+,a)\rightarrow (\mathbb {R}_+, b)$$ . 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 $$\left( \mathbb {R}_+^d, a\right) \rightarrow \left( \mathbb {R}_+^{d'}, b\right) $$ 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.

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.

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).

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].

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.

CONTENTS Introduction § 1. Normal forms of hyperbolic local families § 2. Deformations of germs of vector fields with a single zero eigenvalue § 3. Bifurcation of a separatrix loop of a saddle on the plane References

Normal forms and desingularization Singular points of planar analytic vector fields Local and global theory of linear systems Functional moduli of analytic classification of resonant germs and their applications Global properties of complex polynomial foliations Appendix. First aid Bibliography Index.

Preface. Key to group picture. Participants. Contributors. Relations between Abelian integrals and limit cycles M. Caubergh, R. Roussarie. Topics on singularities and bifurcations of vector fields F. Dumortier, P. de Maesschalck. Recent advances in the analysis of divergence and singularities J. Ecalle. Local bifurcations of limit cycles, Abel equations and Lienard systems J.-P. Francoise. Complexity of computations with Pfaffian and Noetherian functions A. Gabrielov, N. Vorobjov. Hamiltonian bifurcations and local analytic classification V. Gelfreich. Confluence of singular points and Stokes phenomena A. Glutsyuk. Bifurcations of relaxation oscillations J. Guckenheimer. Selected topics in differential equations with real and complex time Y. Ilyashenko. Growth rate of the number of periodic points V.Yu. Kaloshin. Normal forms, bifurcations and finiteness properties of vector fields C. Rousseau. Aspects of planar polynomial vector fields: global versus local, real versus complex, analytic versus algebraic and geometric D. Schlomiuk. Index.

We refine the transformation to smooth normal form for an analytic family of vector fields in the neighbourhood of a saddle-node. This refinement is very powerful and allows us to prove the finite cyclicity of families of graphics (‘ensembles’) occurring inside analytic families of vector fields. In [ZR1] and [ZR2] it is used to prove the finite cyclicity of graphics through a nilpotent singular point of elliptic type. Several examples are presented: lips, graphics with two subsequent lips, graphics with a nilpotent point of elliptic type and a saddle-node. We also discuss the bifurcation diagram of limit cycles for a graphic in the lips.