Unstable manifold expansion for Vlasov-Fokker-Planck equation

We investigate the bifurcation of a homogeneous stationary state of Vlasov-Newton equation in one dimension, in presence of a small dissipation mod-eled by a Fokker-Planck operator. Depending on the relative size of the dissipation and the unstable eigenvalue, we find three different regimes: for a very small dissipa-tion, the system behaves as a pure Vlasov equation; for a strong enough dissipation, the dynamics presents similarities with a standard dissipative bifurcation; in addition, we identify an intermediate regime interpolating between the two previous ones. This work relies on an unstable manifold expansion, performed using Bargman representation for the functions and operators analyzed. The resulting series are estimated with Mellin transform techniques