# Mathematical developments around Hilbert's 16th problem (07w5021)

Arriving in Banff, Alberta Sunday, March 11 and departing Friday March 16, 2007

## Organizers

Christiane Rousseau (Université de Montréal)

## Objectives

The second part of Hilbert's 16 th problem is still open. It is a problem at the confluence of many domains: qualitative theory of ODE, complex foliations, algebraic differential equations and differential Galois theory, analytic theory of differential equations. The purpose of the workshop is to bring together a group of researchers making significant contributions to a variety of domains of differential equations related to Hilbert's 16th problem: among them having made significant contributions to Hilbert's 16th problem itself together with specialists of complex foliations, real and complex dynamical systems, differential Galois theory and resummation techniques.

The focus will be on the following subjects:

1. Singularities of differential equations and complex foliations, and related normal forms. Although the discovery of chaos and wild behaviour of dynamical systems goes back to Poincare, it was known for much longer that very simple differential equations, as the Euler equation, can have divergent solutions. The study of the singularities of differential equations is a wide chapter of dynamical systems. The simplest singularities are studied by the standard technique of normal form. In most of the cases the changes of coordinates to normal form diverge. Among the sources of divergence one finds small divisors and multi-sommability.

A central problem in the study of singularities is the local equivalence problem: how to decide if vector fields or families of vector fields are locally equivalent under a change of coordinates. This amounts to identify representatives of the equivalence classes. A natural approach is to do this through normal forms. It is however known that in many interesting cases, the changes of coordinates to normal form diverge. Moreover in these cases, as soon as there is a potential of divergence, then divergence of the normalizing changes of coordinate is the rule and convergence is the exception. These surprising phenomena can be understood when one unfolds the vector fields. This approach has been proposed for instance by Arnold and Bolibruck. In the case of small divisors it has been studied by Yoccoz and Perez-Marco.

The difficulties of these questions come from the fact that the divergent series occurring in the case of ``small divisors'' problems are not multi-summable. Moreover, in the resonant cases where the one can use the multi-summability techniques introduced for instance by Ecalle or Ramis, these methods do not work when one unfolds the singularity. The geometric methods of the school of Douady, to which belongs Yoccoz, have been developed to study complex dynamics, like the Julia and Mandelbrot sets. They have proved to be been particularly successful in solving certain problems of this kind. They are an essential tool in attacking the local equivalence problem for germs of families of vector fields unfolding a singularity.

2. Bifurcations of differential equations and finite cyclicity problems. As polynomial vector fields of a given degree form analytic families of vector fields a natural approach to Hilbert's 16 th problem is via bifurcations. The workshop will focus on progress in bifurcations of 2-dimensional systems, both for generic systems and inside particular families, for instance families of polynomial vector fields. A special emphasis will be put on the study of singular perturbations and also on the bifurcations of graphics of integrable systems. A subproblem of determining the bifurcation diagram for a given bifurcation occuring on some graphic is the problem of proving the finite cyclicity of some graphics inside special families of analytic vector fields. The study of singularities of families of vector fields and their unfoldings is an essential tool in this program.

3. Algebro-geometric techniques in differential equations. These techniques play a central role in determining the integrability conditions and analysing the bifurcation manifolds. A subproblem of Hilbert's 16th problem, called tangential Hilbert's 16th problem, is concerned with the maximum number and related positions of limit cycles of small perturbations of Hamiltonian vector fields. Far from singularities the problem is then reduced to study the number of zeros of Abelian integrals. However when we approach critical values of the Hamiltonian, not only can there be more limit cycles than the number of zeroes of the Abelian integrals, but the upper bounds on the number of zeroes of Abelian integrals explode. Abelian integrals are studied through the integrable linear Pfaffian systems, which they are solutions of. A solution proposed is to consider these systems in the multidimensional setting where the independent variable and the parameters play the same role and to explore the possibility of getting rid of explodings residues by suitable meromorphic gauge transformation, thus opening the hope to get bounds for the number of zeroes of Abelian integrals even in the neighborhood of critical values. This represents a chapter of complex differential equations and PDE via the Gauss-Manin connection.

A small scientific Committee will prepare the program. The suggested persons on the scientific Committee are Yulij Ilyashenko, Robert Roussarie and Christiane Rousseau. The workshop will start with a review lecture on the main streams of research around Hilbert's 16th problem. The program of the workshop will be built so as to start with the main contributions to the field and significant results in the last few years in the three directions identified above or in any new direction in the field. There will be lectures of 50 minutes and lectures of 25 minutes. Special attention will be given so young people having made important contributions be invited to lecture.

The focus will be on the following subjects:

1. Singularities of differential equations and complex foliations, and related normal forms. Although the discovery of chaos and wild behaviour of dynamical systems goes back to Poincare, it was known for much longer that very simple differential equations, as the Euler equation, can have divergent solutions. The study of the singularities of differential equations is a wide chapter of dynamical systems. The simplest singularities are studied by the standard technique of normal form. In most of the cases the changes of coordinates to normal form diverge. Among the sources of divergence one finds small divisors and multi-sommability.

A central problem in the study of singularities is the local equivalence problem: how to decide if vector fields or families of vector fields are locally equivalent under a change of coordinates. This amounts to identify representatives of the equivalence classes. A natural approach is to do this through normal forms. It is however known that in many interesting cases, the changes of coordinates to normal form diverge. Moreover in these cases, as soon as there is a potential of divergence, then divergence of the normalizing changes of coordinate is the rule and convergence is the exception. These surprising phenomena can be understood when one unfolds the vector fields. This approach has been proposed for instance by Arnold and Bolibruck. In the case of small divisors it has been studied by Yoccoz and Perez-Marco.

The difficulties of these questions come from the fact that the divergent series occurring in the case of ``small divisors'' problems are not multi-summable. Moreover, in the resonant cases where the one can use the multi-summability techniques introduced for instance by Ecalle or Ramis, these methods do not work when one unfolds the singularity. The geometric methods of the school of Douady, to which belongs Yoccoz, have been developed to study complex dynamics, like the Julia and Mandelbrot sets. They have proved to be been particularly successful in solving certain problems of this kind. They are an essential tool in attacking the local equivalence problem for germs of families of vector fields unfolding a singularity.

2. Bifurcations of differential equations and finite cyclicity problems. As polynomial vector fields of a given degree form analytic families of vector fields a natural approach to Hilbert's 16 th problem is via bifurcations. The workshop will focus on progress in bifurcations of 2-dimensional systems, both for generic systems and inside particular families, for instance families of polynomial vector fields. A special emphasis will be put on the study of singular perturbations and also on the bifurcations of graphics of integrable systems. A subproblem of determining the bifurcation diagram for a given bifurcation occuring on some graphic is the problem of proving the finite cyclicity of some graphics inside special families of analytic vector fields. The study of singularities of families of vector fields and their unfoldings is an essential tool in this program.

3. Algebro-geometric techniques in differential equations. These techniques play a central role in determining the integrability conditions and analysing the bifurcation manifolds. A subproblem of Hilbert's 16th problem, called tangential Hilbert's 16th problem, is concerned with the maximum number and related positions of limit cycles of small perturbations of Hamiltonian vector fields. Far from singularities the problem is then reduced to study the number of zeros of Abelian integrals. However when we approach critical values of the Hamiltonian, not only can there be more limit cycles than the number of zeroes of the Abelian integrals, but the upper bounds on the number of zeroes of Abelian integrals explode. Abelian integrals are studied through the integrable linear Pfaffian systems, which they are solutions of. A solution proposed is to consider these systems in the multidimensional setting where the independent variable and the parameters play the same role and to explore the possibility of getting rid of explodings residues by suitable meromorphic gauge transformation, thus opening the hope to get bounds for the number of zeroes of Abelian integrals even in the neighborhood of critical values. This represents a chapter of complex differential equations and PDE via the Gauss-Manin connection.

A small scientific Committee will prepare the program. The suggested persons on the scientific Committee are Yulij Ilyashenko, Robert Roussarie and Christiane Rousseau. The workshop will start with a review lecture on the main streams of research around Hilbert's 16th problem. The program of the workshop will be built so as to start with the main contributions to the field and significant results in the last few years in the three directions identified above or in any new direction in the field. There will be lectures of 50 minutes and lectures of 25 minutes. Special attention will be given so young people having made important contributions be invited to lecture.