Regularization of the Boundary-Saddle-Node Bifurcation
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-sadd...
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2018-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2018/5094878 |
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| author | Xia Liu |
| author_facet | Xia Liu |
| author_sort | Xia Liu |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-f23dc8b4aa9e44a0982689f644d16bf9 |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Advances in Mathematical Physics |
| spelling | doaj-art-f23dc8b4aa9e44a0982689f644d16bf92025-02-03T01:25:42ZengWileyAdvances in Mathematical Physics1687-91201687-91392018-01-01201810.1155/2018/50948785094878Regularization of the Boundary-Saddle-Node BifurcationXia Liu0School of Mathematics and Physics, Hebei GEO University, Shijiazhuang 050031, ChinaIn 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.http://dx.doi.org/10.1155/2018/5094878 |
| spellingShingle | Xia Liu Regularization of the Boundary-Saddle-Node Bifurcation Advances in Mathematical Physics |
| title | Regularization of the Boundary-Saddle-Node Bifurcation |
| title_full | Regularization of the Boundary-Saddle-Node Bifurcation |
| title_fullStr | Regularization of the Boundary-Saddle-Node Bifurcation |
| title_full_unstemmed | Regularization of the Boundary-Saddle-Node Bifurcation |
| title_short | Regularization of the Boundary-Saddle-Node Bifurcation |
| title_sort | regularization of the boundary saddle node bifurcation |
| url | http://dx.doi.org/10.1155/2018/5094878 |
| work_keys_str_mv | AT xialiu regularizationoftheboundarysaddlenodebifurcation |