Generic bifurcations of planar Filippov systems via geometric singular perturbations (Q657668)

The paper presents a description of the typical phase portraits for discontinuous planar systems of ordinary differential equations. The local behaviour in a vicinity of a singular point on the corresponding discontinuity set \(\Sigma\) is studied via regularization and singular perturbation theory. The first result concerns the case when the vector field is transversal to \(\Sigma\) from one side of it and has either a hyperbolic saddle or a hyperbolic focus or a \(\Sigma\)-cusp point crossing \(\Sigma\) from the other side. Then, it is shown that the bifurcation can be completely described by a suitable singular perturbation problem. Next, the case when both vector fields have \(\Sigma\)-fold points is investigated. Similarly, a description of the generic bifurcations is presented.