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## ON NON-SMOOTH VECTOR FIELDS HAVING A TORUS OR A SPHERE AS THE SLIDING MANIFOLD

### Citations

200 |
Filippov, Differential equations with discontinuous righthand side
- F
- 1985
(Show Context)
Citation Context ...phenomenon has been studied specially on relay control systems and systems with dry friction [3, 7]. The mathematical formalization of the theory of discontinuous systems was made precise by Filippov =-=[5]-=- and Teixeira-Sotomayor [12, 10], which classified and distinguished three types of regions on the discontinuity manifold (sewing region, sliding region and escape region). These regions fully describ... |

63 |
Nonsmooth Mechanics — Models, Dynamics and Control
- Brogliato
- 1999
(Show Context)
Citation Context ...13371-9 and 2012/06879-1. 1 ar X iv :1 20 7. 03 97 v1s[ ma th. DS ]s2 J uls20 12 2 RICARDO MIRANDA MARTINS (separated by tangency curves). Inelastic vector fields are widely used in mechanical models =-=[2]-=-. This paper is outlined as follows: in Section 2, we introduce the concept of nonsmooth vector fields. In Section 3 we discuss inelastic vector fields. In Section 4 we state the main results. In Sect... |

21 |
Stability conditions for discontinuous vector fields
- Teixeira
- 1990
(Show Context)
Citation Context ... specially on relay control systems and systems with dry friction [3, 7]. The mathematical formalization of the theory of discontinuous systems was made precise by Filippov [5] and Teixeira-Sotomayor =-=[12, 10]-=-, which classified and distinguished three types of regions on the discontinuity manifold (sewing region, sliding region and escape region). These regions fully describes what can happen on such manif... |

18 | Generic bifurcations of low codimension of planar Filippov systems
- Guardia, Seara, et al.
(Show Context)
Citation Context ...portrait. The boundary of such regions is called the discontinuity manifold [8]. There are a lot of research being made about the local behavior of non-smooth systems, near the discontinuity manifold =-=[6]-=-. An interesting phenomenon that occurs frequently is the sliding motion, when the trajectories on both sides of the discontinuity manifold slides over the manifold after the meeting, and there remain... |

13 |
Positive definite matrices
- Johnson
- 1970
(Show Context)
Citation Context ...y points over the sphere. SLIDING VECTOR FIELD OVER TORI AND SPHERES 5 Proof. We know by the discussion of the last paragraph that if Q = A + At is positive definite, then there are no tangencies. In =-=[9]-=- is shown that a matrix L is positive (negative) definite if, and only if, L + Lt is positive (negative) definite. This complete the proof. Lemma 3. Suppose that A is a negative definite matrix. The... |

11 |
Regularization of discontinuous vector fields,”
- Sotomayor, Teixeira
- 1996
(Show Context)
Citation Context ... specially on relay control systems and systems with dry friction [3, 7]. The mathematical formalization of the theory of discontinuous systems was made precise by Filippov [5] and Teixeira-Sotomayor =-=[12, 10]-=-, which classified and distinguished three types of regions on the discontinuity manifold (sewing region, sliding region and escape region). These regions fully describes what can happen on such manif... |

9 | Teixeira singularities in 3D switched feedback control systems
- Colombo, Bernardo, et al.
- 2010
(Show Context)
Citation Context ...s the sliding motion, when the trajectories on both sides of the discontinuity manifold slides over the manifold after the meeting, and there remains until reaching the boundary of the sliding region =-=[4, 11]-=-. This phenomenon has been studied specially on relay control systems and systems with dry friction [3, 7]. The mathematical formalization of the theory of discontinuous systems was made precise by Fi... |

8 |
Discontinuity geometry for an impact oscillator
- Chillingworth
- 2002
(Show Context)
Citation Context ...ifold after the meeting, and there remains until reaching the boundary of the sliding region [4, 11]. This phenomenon has been studied specially on relay control systems and systems with dry friction =-=[3, 7]-=-. The mathematical formalization of the theory of discontinuous systems was made precise by Filippov [5] and Teixeira-Sotomayor [12, 10], which classified and distinguished three types of regions on t... |

8 |
Perturbation theory for non-smooth systems
- Teixeira
- 2009
(Show Context)
Citation Context ...s the sliding motion, when the trajectories on both sides of the discontinuity manifold slides over the manifold after the meeting, and there remains until reaching the boundary of the sliding region =-=[4, 11]-=-. This phenomenon has been studied specially on relay control systems and systems with dry friction [3, 7]. The mathematical formalization of the theory of discontinuous systems was made precise by Fi... |

4 |
Study of singularities in nonsmooth dynamical systems via singular perturbation
- Llibre, Silva, et al.
(Show Context)
Citation Context ... the non-smooth systems are described by systems of ODEs in such a way that each system is defined in a region of the phase portrait. The boundary of such regions is called the discontinuity manifold =-=[8]-=-. There are a lot of research being made about the local behavior of non-smooth systems, near the discontinuity manifold [6]. An interesting phenomenon that occurs frequently is the sliding motion, wh... |

3 |
On the Mechanics of Large Inelastic Deformations: Kinematics and Constitutive Modeling,"
- Zbib
- 1993
(Show Context)
Citation Context ...regular value and consider the manifold M = h−1({0}). Two vector fields X,Y are inelastic over M if (2) (Xh)(x) = −(Y h)(x) for all x ∈M . Inelastic vector fields appears naturally in mechanic models =-=[13]-=- and in reversible non-smooth models. A planar non-smooth vector field Z = (X,Y ) with discontinuity set Σ ⊂ R2 is said to be reversible with respect to an involution ϕ : R2 → R2 with Fix(ϕ) ⊂ Σ, wher... |

1 |
Coupled systems of non-smooth differential equations
- Jacquemard, Tonon
- 2012
(Show Context)
Citation Context ...ifold after the meeting, and there remains until reaching the boundary of the sliding region [4, 11]. This phenomenon has been studied specially on relay control systems and systems with dry friction =-=[3, 7]-=-. The mathematical formalization of the theory of discontinuous systems was made precise by Filippov [5] and Teixeira-Sotomayor [12, 10], which classified and distinguished three types of regions on t... |