Polynomial normal forms with exponentially small remainder for analytic vector fields

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

Robust normal forms for saddles of analytic vector fields

The aim of this paper is to introduce a technique for describing trajectories of systems of ordinary differential equations (ODEs) passing near saddle-fixed points. In contrast to classical linearization techniques, the methods of this paper allow for perturbations of the underlying vector fields. This robustness is vital when modelling systems containing small uncertainties, and in the develop...

Polynomial Normal Forms for Vector Fields on R3

The present paper is devoted to studying a class of smoothly (C∞) finitely determined vector fields on R3. Given any such generic local system of the form ẋ = Ax + · · · , where A is a 3 × 3 matrix, we find the minimal possible number i(A) such that the vector field is i(A)-jet determined, and we find the number μ(A) of moduli in the C∞ classification. We also give a list of the simplest normal...

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...