Title: Structural bifurcation of 2-D incompressible flows

Authors: Michael Ghil, Tian Ma and Shouhong Wang

Issue: Volume 50 (2001), Issue 1, 159-180

Abstract: 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.