Structural bifurcation of 2-D incompressible flows

We study in this article the structural bifurcation of divergence-free vector fields on a two-dimensional (2-D) compact manifold M. We prove that, for a one-parameter family of divergence-free vector fields u ((.), t) structural bifurcation - i.e. change in their topological-equivalence class-occurs at to if u((.), t(0)) has a degenerate singular point x(0) is an element of partial derivativeM such that partial derivativeu(x(0), t(0))/ partial derivativet not equal 0. Careful analysis of the trajectories allows us to give a complete classification of the orbit structure of u (x, t) near (x(0), t(0)). This article is part of a program to develop a geometric theory for the Lagrangian dynamics of 2-D incompressible fluid flows.