Robust normal forms for saddles of analytic vector fields

On the flat remainder in normal forms of families of analytic planar saddles

We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter.

Normal Forms of Vector Fields on Poisson Manifolds

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

Normal Forms of Analytic Perturbations of Quasihomogeneous Vector Fields: Rigidity, Invariant Analytic Sets and Exponentially Small Approximation

Normal forms with exponentially small remainder : application to homoclinic connections for the reversible 0 2+ i resonance

In this note we explain how the normal form theorem established in [2] for analytic vector fields with a semisimple linearization enables to prove the existence of homoclinic connections to exponentially small periodic orbits for reversible analytic vector fields admitting a 0iω resonance where the linearization is precisely not semi simple.

First Integrals and Normal Forms for Germs of Analytic Vector Fields

For a germ of analytic vector fields, the existence of first integrals, resonance and the convergence of normalization transforming the vector field to a normal form are closely related. In this paper we first provide a link between the number of first integrals and the resonant relations for a quasi-periodic vector field, which generalizes one of the Poincaré’s classical results [18] on autono...