Jean-Pierre Francoise

Successive derivatives of a first return map, application to the study of quadratic vector fields

We provide an algorithm to compute the first non-zero derivative of the return map r ↦ L(r, e) of a planar vector field which is a polynomial perturbation of a linear center. It yields a new method for finding the centre conditions in the case of a homogeneous perturbative part.

Keeping track of limit cycles

This article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields. It introduces the notion of limit periodic sets and shows that the Hilberts 16th problem can be reduced to the existence of a uniform bound on its cyclicity. It also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the Noetherian property of the Bautin ideal.

Center conditions, compositions of polynomials and moments on algebraic curves

We consider an Abel equation

Center conditions III: Parametric and model center problems

(*)

A completely integrable Hamiltonian system

y^{\prime}=p(x)y^2+q(x)y^3

Periodic Motions Galore: How to Modify Nonlinear Evolution Equations so that They Feature a Lot of Periodic Solutions

with

Generalized Moments, Center-Focus Conditions, and Compositions of Polynomials

p(x)

Canonical partition functions of Hamiltonian systems and the stationary phase formula

,

Algebraic linearization of dynamics of Calogero type for any Coxeter group

q(x)