Yulij Ilyashenko

Normal Forms, Bifurcations and Finiteness Problems in Differential Equations

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.

Invisible parts of attractors

This paper deals with attractors of generic dynamical systems. We introduce the notion of e-invisible set, which is an open set of the phase space in which almost all orbits spend on average a fraction of time no greater than e. For extraordinarily small values of e (say, smaller than 2−100), these are large neighbourhoods of some parts of the attractors in the phase space which an observer virtually never sees when following a generic orbit.For any n ≥ 100, we construct a set Qn in the space of skew products over a solenoid with the fibre a circle having the following properties. Any map from Qn is a structurally stable diffeomorphism; the Lipschitz constants of the map and its inverse are no greater than L (where L is a universal constant that does not depend on n, say L < 100). Moreover, any map from Qn has a 2−n-invisible part of its attractor, whose size is comparable to that of the whole attractor. The set Qn is a ball of radius O(n−2) in the space of skew products with the C1 metric. It consists of structurally stable skew products.Small perturbations of these skew products in the space of all diffeomorphisms still have attractors with the same properties. Thus for all such perturbations, a sizable portion of the attractor is almost never visited by generic orbits and is practically never seen by the observer.

Normal forms near a saddle-node and applications to finite cyclicity of graphics

Limburgs Univ Ctr, B-3590 Diepenbeek, Belgium. Cornell Univ, Dept Math, Ithaca, NY 14853 USA. State & Independent Moscow Univ, VA Steklov Math Inst, Moscow 117333, Russia. Univ Montreal, Dept Math & Stat, Montreal, PQ H3C 3J7, Canada. Univ Montreal, CRM, Montreal, PQ H3C 3J7, Canada.Dumortier, F, Limburgs Univ Ctr, Univ Campus, B-3590 Diepenbeek, Belgium.

Hölder properties of perturbed skew products and Fubini regained

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 Gorodetskis 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.

Towards the General Theory of Global Planar Bifurcations

This is an outline of a theory to be created, as it was seen in April 2015. An addendnum to the proofs at the end of the chapter describes the recent developments.

Local and global theory of linear systems

Analysis of holomorphic vector fields and analytic foliations beyond the local theory exposed in Chapter I, is very difficult in more than two dimensions. Perhaps the only case where such a study is possible, both locally and globally, is that of (nonautonomous) linear systems. These systems exist on a rather special type of holomorphic manifolds, holomorphic vector bundles. The latter are “locally cylindric manifolds” made of cylinders (Cartesian products) U ×Cn, U ⊆ C in the same way the manifolds are made of locally Euclidean charts. In this section we develop local and global theory of linear systems and their singularities.