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.