Abstract
We study in this article the structural bifurcation of divergence-free vector fields on a two-dimensional (2-D) com- pact manifold M. We prove that, for a one-parameter family of divergence-free vector fields uN;tO structural bifurcation— i.e., change in their topological-equivalence class—occurs at t0 if uN;t 0O has a degenerate singular point x0 2 @M such that @uNx0;t 0O=@t 6E 0. Careful analysis of the trajectories allows us to give a complete classification of the orbit structure ofuNx;tO nearNx0;t 0O. This article is part of a program to develop a geo- metric theory for the Lagrangian dynamics of 2-D incompressible fluid flows.
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