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## GENERIC BIFURCATION OF CERTAIN PIECEWISE SMOOTH VECTOR FIELDS.

### Citations

782 |
Differential Equations with Discontinuous Right-Hand Side
- Filippov
- 1964
(Show Context)
Citation Context ...typical singularities of NSDS in the plane are discussed. We study in this setting a set of typical bifurcations which are not found in smooth systems. We focus our attention on Filippov systems (see =-=[7]-=-), which are systems modelled by ordinary differential equations with discontinuous righthand sides. It is well known that many of these models (see for instance [2] and [3]) occur in generic k−parame... |

64 |
Canard cycles and center manifolds,
- Dumortier, Roussarie
- 1996
(Show Context)
Citation Context ...Canard cycle of kind II. Figure 9. Canard cycle of kind III. Remark 1. The expression “canard” is used here because these orbits are limit periodic sets of singular perturbation problems (see [4] and =-=[6]-=-). Definition 3. Consider Z ∈ Ωr. A closed path ∆ is a Σ−graph if it is a union of equilibria, pseudo equilibria, tangential singularities of Z and orbit−arcs of Z joining these points in such a way t... |

49 | One-parameter bifurcations in planar Filippov system”.
- Kuznetsov, Rinaldi, et al.
- 2003
(Show Context)
Citation Context ...ns and related problems involving or not sliding regions are studied in papers like [5] and [8]. The classification of codimension 1 local and some global bifurcations for planar systems was given in =-=[11]-=-. In [9] is shown how to construct the homeomorphisms which lead to equivalences between two non−smooth systems when the discontinuity set is a planar smooth curve. In that work codimension two singul... |

18 | Generic bifurcations of low codimension of planar Filippov systems
- Guardia, Seara, et al.
(Show Context)
Citation Context ...lated problems involving or not sliding regions are studied in papers like [5] and [8]. The classification of codimension 1 local and some global bifurcations for planar systems was given in [11]. In =-=[9]-=- is shown how to construct the homeomorphisms which lead to equivalences between two non−smooth systems when the discontinuity set is a planar smooth curve. In that work codimension two singularities ... |

11 |
Regularization of discontinuous vector fields,”
- Sotomayor, Teixeira
- 1996
(Show Context)
Citation Context ...ions of q̇ = Z(q), which has, in general, discontinuous right−hand side. The basic results of differential equations, in this context, were stated by Filippov in [7]. Related theories can be found in =-=[10, 12, 14]-=-. Definition 1. Two non−smooth vector fields Z, Z̃ ∈ Ωr(K, f) defined in open sets U, Ũ ∈ K and with switching manifolds Σ ⊂ U and Σ̃ ⊂ Ũ respectively are Σ−equivalent if there exists an orientation... |

9 |
Two-parameter discontinuity-induced bifurcations of limit cycles: Classification and open problems”.
- Bernardo, Champneys, et al.
- 2006
(Show Context)
Citation Context ...n on Filippov systems (see [7]), which are systems modelled by ordinary differential equations with discontinuous righthand sides. It is well known that many of these models (see for instance [2] and =-=[3]-=-) occur in generic k−parameter families and therefore they typically undergo generic codimension k bifurcations. Many authors have contributed to the study of Filippov systems (see for instance [7] an... |

8 |
Structurally stable systems
- Andronov, Pontryagin
- 1937
(Show Context)
Citation Context ... paper we focus on certain unstable non−smooth vector fields within a generic context. The framework in which we shall pursue these unstable systems is sometimes called generic bifurcation theory. In =-=[1]-=- the concept of kth−order structural stability is presented; in a local approach such setting gives rise to the notion of a codimension k singularity. Observe that, so far, bifurcation and normal form... |

8 |
Generic bifurcation in manifolds with boundary
- Teixeira
- 1977
(Show Context)
Citation Context ...on, non−smooth vector field. 1 2 C.A. BUZZI, T. DE CARVALHO AND M.A. TEIXEIRA [10]). One of the starting points for a systematic approach in the geometric and qualitative analysis of NSDS is the work =-=[13]-=-, of M. A Teixeira, on smooth systems in 2−dimensional manifolds with boundary. The generic singularities that appear in NSDS, as far as we know, were first studied in [14]. Bifurcations and related p... |

8 |
Perturbation theory for non-smooth systems
- Teixeira
- 2009
(Show Context)
Citation Context ...bal bifurcations. These bifurcations, that also appear in the present work, called ST−bifurcations, are characterized by the connection between a saddle critical point and a tangency singularity. See =-=[15]-=- for a survey on NSDS and references there in. Those papers give the necessary basis for the development of our approach. The specific topic addressed in this paper is the characterization of specific... |

7 |
Roughness of a discontinuous system
- Kozlova
- 1984
(Show Context)
Citation Context ...lassification. Primary 34A36, 37G10, 37G05. Key words and phrases. Fold−Saddle singularity, canard, limit cycle, bifurcation, non−smooth vector field. 1 2 C.A. BUZZI, T. DE CARVALHO AND M.A. TEIXEIRA =-=[10]-=-). One of the starting points for a systematic approach in the geometric and qualitative analysis of NSDS is the work [13], of M. A Teixeira, on smooth systems in 2−dimensional manifolds with boundary... |

5 |
Non-smooth pitchfork bifurcations, Discrete Contin
- Glendinning
(Show Context)
Citation Context ...ry. The generic singularities that appear in NSDS, as far as we know, were first studied in [14]. Bifurcations and related problems involving or not sliding regions are studied in papers like [5] and =-=[8]-=-. The classification of codimension 1 local and some global bifurcations for planar systems was given in [11]. In [9] is shown how to construct the homeomorphisms which lead to equivalences between tw... |

5 |
Generic singularities of discontinuous vector
- Teixeira
- 1981
(Show Context)
Citation Context ...nalysis of NSDS is the work [13], of M. A Teixeira, on smooth systems in 2−dimensional manifolds with boundary. The generic singularities that appear in NSDS, as far as we know, were first studied in =-=[14]-=-. Bifurcations and related problems involving or not sliding regions are studied in papers like [5] and [8]. The classification of codimension 1 local and some global bifurcations for planar systems w... |

3 |
Center-focus and isochronous center problems for discontinuous differential equations, Discrete Contin
- Coll, Gasull, et al.
(Show Context)
Citation Context ...h boundary. The generic singularities that appear in NSDS, as far as we know, were first studied in [14]. Bifurcations and related problems involving or not sliding regions are studied in papers like =-=[5]-=- and [8]. The classification of codimension 1 local and some global bifurcations for planar systems was given in [11]. In [9] is shown how to construct the homeomorphisms which lead to equivalences be... |

3 |
Vector fields near the boundary of a manifold, Vestinik Moskovskogo Universiteta. Matematika 27
- Vishik
- 1972
(Show Context)
Citation Context ...the Cr−topology. In this universe a Σ−fold point has codimension zero. It is possible to consider f(x, y) = y and the following generic normal forms X0(x, y) = (α1, β1x) with α1 = ±1 and β1 = ±1 (see =-=[16]-=-, Theorem 2). Let Y0 be a smooth vector field defined in Σ−. Assume that Y0 has a hyperbolic saddle point SY0 on Σ and that the eigenspaces of DY0(SY0) are transverse to Σ at SY0 . We denote the set o... |

1 |
Closed poly−trajectories and Poincaré Index of non-Smooth vector fields on the plane
- Buzzi, Carvalho, et al.
(Show Context)
Citation Context ...gure 8. Canard cycle of kind II. Figure 9. Canard cycle of kind III. Remark 1. The expression “canard” is used here because these orbits are limit periodic sets of singular perturbation problems (see =-=[4]-=- and [6]). Definition 3. Consider Z ∈ Ωr. A closed path ∆ is a Σ−graph if it is a union of equilibria, pseudo equilibria, tangential singularities of Z and orbit−arcs of Z joining these points in such... |