We present a numerical analysis of a class of radial-anisotropic collisonless spherical stellar systems in order to understand the physical ori¬gin of the radial-orbit instability. This work is motivated by new analytical results based on the symplectic formulation of a generalised energy varia¬tionnal technique. We have obtained a first confirmation that the stability of such systems is governed by their aptitude to receive a certain class of perturbations.