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.
In 1970 Donald Saari conjectured that in the N-body 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: McCord-the 3-body equal-mass case, Moeckel-the general 3-body case, Llibre and Pina-both previous cases. In collaboration with Manuele Santoprete and Ernesto Perez-Chavela, I have proved the collinear N-body 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.
The cocoon bifurcation is a set of rich bifurcation phenomena numerically observed in the three-dimensional ODE system for travelling waves of the Kuramoto-Sivashinsky equation. In this talk we present the "cusp-transverse heteroclinic chain" as organizing center of a basic part of the cocoon bifurcation in the more general setting of reversible vector fields on R3.
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.
We give resurgence properties and derive analytic invariants for some one-parameter complex analytic dynamical systems. Parabolic implosions of (one-dimensional) diffeomorphisms and saddle node bifurcations for vector fields in the complex plane will, in particular, be considered.
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 two-parameter 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, Two-dimensional Fuchsian systems and the Chebyshev property. J. Differential Equations (1) 191(2003), 105-120.
By gluing foliated complex manifolds, we deduce analytic normal forms for germs of real or complex foliations or planar vector fields.
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.
Let us denote respectively by QH and QR the Hamiltonian class and reversible class of quadratic integrable systems. There are several topological types for systems belonging to QHÇQR. One of them is the case where the corresponding system has two heteroclinic loops with 2 saddles (2-saddle-loops), sharing one saddle-connection, which is a line segment (the other saddle-connections 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 2-saddle-loops 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.
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 2-dimensional 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.
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.
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 C2 (or R2). 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).
I shall talk about the finite cyclicity of nilpotent graphics of HH type and applications to Hilbert's 16th problem.