Structural bifurcation of 2-D incompressible flows Michael GhilTian MaShouhong Wang 34D35Q3558F7686A10divergence-free vector fieldsLagrangian dynamicsincompressible flowsstructural stabilitystructural bifurcationsingularity classification 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 $u(\cdot,t)$ structural bifurcation --- i.e., change in their topological-equivalence class --- occurs at $t_0$ if $u(\cdot,t_0)$ has a degenerate singular point $x_0 \in \partial M$ such that $\partial u(x_0,t_0)/\partial t \not= 0$. 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. Indiana University Mathematics Journal 2001 text pdf 10.1512/iumj.2001.50.2183 10.1512/iumj.2001.50.2183 en Indiana Univ. Math. J. 50 (2001) 159 - 180 state-of-the-art mathematics http://iumj.org/access/