Weinkauf, T.Theisel, H.Hege, H.-C.Seidel, H.-P.2015-02-212015-02-2120061467-8659https://doi.org/10.1111/j.1467-8659.2006.00980.xIn this paper we extract and visualize the topological skeleton of two-parameter-dependent vector fields. This kind of vector data depends on two parameter dimensions, for instance physical time and a scale parameter. We show that two important classes of local bifurcations - fold and Hopf bifurcations - build line structures for which we present an approach to extract them. Furthermore we show that new kinds of structurally stable local bifurcations exist for this data, namely fold-fold and Hopf-fold bifurcations. We present a complete classification of them. We apply our topological extraction method to analyze a number of two-parameter-dependent vector fields with different physical interpretations of the two additional dimensions.Categories and Subject Descriptors (according to ACM CCS): I.3.3 [Computer Graphics]: Line and Curve Generation I.3.3 [Computer Graphics]: Picture/Image Generation10.1111/j.1467-8659.2006.00980.x607-616