Regularization of the Boundary-Saddle-Node Bifurcation

Abstract EN

In this paper we treat a particular class of planar Filippov systems which consist of two smooth systems that are separated by a discontinuity boundary.

In such systems one vector field undergoes a saddle-node bifurcation while the other vector field is transversal to the boundary.

The boundary-saddle-node (BSN) bifurcation occurs at a critical value when the saddle-node point is located on the discontinuity boundary.

We derive a local topological normal form for the BSN bifurcation and study its local dynamics by applying the classical Filippov’s convex method and a novel regularization approach.

In fact, by the regularization approach a given Filippov system is approximated by a piecewise-smooth continuous system.

Moreover, the regularization process produces a singular perturbation problem where the original discontinuous set becomes a center manifold.

Thus, the regularization enables us to make use of the established theories for continuous systems and slow-fast systems to study the local behavior around the BSN bifurcation.