Regularization of the Boundary‐Saddle‐Node Bifurcation

Xia Liu

Advances in Mathematical Physics, 2018, vol. 2018, issue 1

Abstract: 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.

Date: 2018
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https://doi.org/10.1155/2018/5094878

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2018:y:2018:i:1:n:5094878

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