


SS7  Systèmes dynamiques / SS7  Dynamical systems Org: R. Roussarie (Dijon) et/and C. Rousseau (Montréal)
 ALEXANDER BRUDNYI, University of Calgary
On the center problem for ODEs

In the talk we describe a general approach to the classical center
problem for ODEs and its connection with some results in Algebra and
Complex Analysis.
 FLORIN DIACU, Department of Mathematics and Statistics, University of
Victoria
Saari's Conjecture

In 1970 Donald Saari conjectured that in the Nbody problem of
celestial mechanics every solution with constant moment of inertia is
a relative equilibrium. There have been many attempts to prove the
conjecture, but all of them failed. Recently several researchers have
attacked various particular cases: McCordthe 3body equalmass
case, Moeckelthe general 3body case, Llibre and Pinaboth
previous cases. In collaboration with Manuele Santoprete and Ernesto
PerezChavela, I have proved the collinear Nbody case for any
potential in which the force depends on the mutual distances. This
result and several related ones are the subject of this talk.
 FREDDY DUMORTIER, Limburgs Universitair Centrum, Universitaire Campus
Diepenbeek B, 3590 Belgium
Rigorous approach to cocoon bifurcation in 3dimensional
reversible vector fields

The cocoon bifurcation is a set of rich bifurcation phenomena
numerically observed in the threedimensional ODE system for
travelling waves of the KuramotoSivashinsky equation. In this talk we
present the "cusptransverse heteroclinic chain" as organizing
center of a basic part of the cocoon bifurcation in the more general
setting of reversible vector fields on R^{3}.
We also discuss the relation with a heteroclinic cycle called the
reversible Bykov cycle. The talk is based on results obtained in
collaboration with S. Ibanez and H. Kokubu.
 FRÉDÉRIC FAUVET, IRMAUniversité de Strasbourg 1, 7 rue Descartes, F67084
Strasbourg Cedex
Resurgence in some oneparameter bifurcations

We give resurgence properties and derive analytic invariants for some
oneparameter complex analytic dynamical systems. Parabolic
implosions of (onedimensional) diffeomorphisms and saddle node
bifurcations for vector fields in the complex plane will, in
particular, be considered.
 LUBOMIR GAVRILOV, Université Paul Sabatier, 118 route de Narbonne, 31062
Toulouse Cedex 04
Twodimensional Fuchsian systems and the Chebyshev property

Let ( x(t),y(t) )^{T} be a solution of a Fuchsian system
of order two with three singular points. The vector space of
functions of the form P(t) x(t) + Q(t) y(t), where P, Q are real
polynomials, has a natural filtration of vector spaces, according to
the asymptotic behaviour of the functions at infinity. We describe a
twoparameter class of Fuchsian systems, for which the corresponding
vector spaces obey the Chebyshev property (the maximal number of
isolated zeroes of each function is less than the dimension of the
vector space). Up to now, only a few particular systems were known to
possess such a nonoscillation property. It is remarkable that most of
these systems are of the type studied in the present paper. We apply
our results in estimating the number of limit cycles that appear after
small polynomial perturbations of several quadratic or cubic
Hamiltonian systems in the plane.
For a complete text see:
Lubomir Gavrilov and Iliya D. Iliev, Twodimensional Fuchsian
systems and the Chebyshev property. J. Differential Equations (1)
191(2003), 105120.
 FRANK LORAY, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes
Cedex
Analytic normal forms for planar vector fields by gluing
complex manifolds

By gluing foliated complex manifolds, we deduce analytic normal forms
for germs of real or complex foliations or planar vector fields.
 PAVAO MARDESIC, Université de Bourgogne, Dijon
Le diagramme de bifurcation de la période des centres
quadratiques

Dans ce travail en collaboration avec D. Marín et
J. Villadelprat, nous étudions le diagramme de bifurcation des
points critiques de la période pour les centres quadratiques.
Nous concentrons nos efforts sur la strate la plus intéressante des
systèmes réversibles et donnons une description assez complète
de la partie du diagramme de bifurcation correspondant au bifurcations
des points critiques au niveau du polycycle bordant le bassin du
centre.
 ROBERT ROUSSARIE, Université de Bourgogne
The cyclicity problem for nongeneric quadratic Hamiltonian
systems

Let us denote respectively by Q^{H} and Q^{R} the Hamiltonian class
and reversible class of quadratic integrable systems. There are
several topological types for systems belonging to Q^{H}ÇQ^{R}. One
of them is the case where the corresponding system has two
heteroclinic loops with 2 saddles (2saddleloops), sharing
one saddleconnection, which is a line segment (the other
saddleconnections form an ellipse).
In a recent work in collaboration with Chengzhi Li, we prove that the
maximal number of limit cycles which bifurcate from the union of the
2saddleloops with respect to quadratic perturbations is two,
in the reversible direction. This means that every bifurcating
limit cycle is related to a zero of Abelian integral. We also describe
the corresponding bifurcation diagram. This diagram cannot be
deduced from the bifurcation diagram for Abelian integral zeros. It
must be noticed that the bifurcation diagram is not completely
established, up to now, outside a conic sector around the reversible
direction in the parameter space.
 CHRISTIANE ROUSSEAU, Université de Montréal
Modulus of analytic classification of a family unfolding a
generic resonant diffeomorphism

We present here a complete modulus of analytic classification for a
germ of generic family unfolding a generic resonant diffeomorphism. We
apply it to derive a complete modulus for a family unfolding a generic
resonant saddle of a 2dimensional vector field and discuss the
geometry of the systems in the family. We discuss the geometry of the
unfolded diffeomorphism and unfolded vector field with a resonant
saddle. From this geometry we explain the meaning of the divergence of
the normalizing series.
 DANA SCHLOMIUK, Université de Montréal
On Hilbert's 16th problem for quadratic systems

In this lecture we shall report on recent progress on this problem.
New results show the major role played by the symmetric (reversible)
systems with center in this problem and the significant part played by
the perturbations of the most degenerate ones among them. Global
concepts as well as theorems of a truly global character are used in
obtaining these results.
 LOÏC TEYSSIER, Université de Lille I, Cité Scientifique, F59655
Villeneuve d'Ascq Cedex
Homological equation of vector fields: from analytic moduli
to limit cycles

Our objects are germs at (0,0) of holomorphic (or real analytic)
vector fields Z. We first address the analytic classification of
these vector fields having an elementary singularity. We reduce the
problem to solving two homological equations associated to Z (that
is, equations of the form Z(F)=G). They are solved by integrating
G along paths tangent to Z, so that the obstructions to the
holomorphy of F are located in the asymptotic cycles of the
underlying foliation. As an application we give a "natural"
integral representation of the invariants of Martinet and Ramis
associated to resonant foliations, yielding many explicit examples of
nonconjugated foliations.
Another application of this method allows us to give analytic normal
forms for affine actions of the plane C^{2} (or
R^{2}). We show that any couple (Z,Y) of holomorphic
vector fields satisfying [Z,Y]=Y is conjugated to a unique model
([(Z)\tilde],[(Y)\tilde]) where [(Z)\tilde] is polynomial and
[(Y)\tilde] is global but multivaluated away from (0,0).
 HUAIPING ZHU, York
Nilpotent graphics and Hilbert's 16th problem for quadratic systems

I shall talk about the finite cyclicity of nilpotent graphics of HH
type and applications to Hilbert's 16th problem.

