Structural Bifurcation of DivergenceFree Vector Fields near Nonsimple Degenerate Points with Symmetry 
Ali Deliceoglu,Deniz Bozkurt 
Keywords:Structural stability, divergencefree vector field, Lagrangian dynamics, structural bifurcation, singularity classification 
Abstract: 
In this study, topological features of an incompressible twodimen\sional flow far from any boundaries is considered. A rigorous theory has been developed for degenerate streamline patterns and their bifurcation. The homotopy invariance of the index is used to simplify the differential equations of fluid flows which are parameter families of divergencefree vector fields. When the degenerate flow pattern is perturbed slightly, a structural bifurcation for flows with symmetry is obtained. We give possible flow structures near a bifurcation point. A flow pattern is found where a degenerate cusp point appears on the xaxis. Moreover, we also show that bifurcation of the flow structure near a nonsimple degenerate critical point with double symmetry is generic away from boundaries. Finally, we give an application of the degenerate flow patterns emerging when index 0 and 2 in a double lid driven cavity and in two dimensional peristaltic flow. 



