Interior Structural Bifurcation of 2D Symmetric Incompressible Flows

The structural bifurcation of a 2D divergence free vector field \begin{document}$ \mathbf{u}(\cdot, t) $\end{document} when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} of zero index has been studied by Ma and Wang [ 23 ]. Although in the class of divergence free fields which undergo a local bifurcation around a singular point, the ones with index zero singular points are generic, this class excludes some important families of symmetric flows. In particular, when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} is anti-symmetric with respect to \begin{document}$ \mathbf{x}_0 $\end{document} , or symmetric with respect to the axis located on \begin{document}$ \mathbf{x}_0 $\end{document} and normal to the unique eigendirection of the Jacobian \begin{document}$ D\mathbf{u}(\cdot, t_0) $\end{document} , the vector field must have index 1 or -1 at the singular point. Thus we study the structural bifurcation when \begin{document}$ \mathbf{u}(\cdot, t_0) $\end{document} has an interior isolated singular point \begin{document}$ \mathbf{x}_0 $\end{document} with index -1, 1. In particular, we show that if such a vector field with its acceleration at \begin{document}$ t_0 $\end{document} both satisfy the aforementioned symmetries then generically the flow will undergo a local bifurcation. Under these generic conditions, we rigorously prove the existence of flow patterns such as pairs of co-rotating vortices and double saddle connections. We also present numerical evidence of the Stokes flow in a rectangular cavity showing that the bifurcation scenarios we present are indeed realizable.