American Institute of Mathematical Sciences

Advanced
April  1997, 3(2): 235-241. doi: 10.3934/dcds.1997.3.235

Regularization of discontinuous vector fields in dimension three

Jaume Llibre 1, and  Marco Antonio Teixeira 2,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2. 

Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P., Brazil

Received  September 1996 Published  January 1997

In this paper vector fields around the origin in dimension three which are approximations of discontinuous ones are studied. In a former work of Sotomayor and Teixeira [6] it is shown, via regularization, that Filippov's conditions are the natural ones to extend the orbit solutions through the discontinuity set for vector fields in dimension two. In this paper we show that this is also the case for discontinuous vector fields in dimension three. Moreover, we analyse the qualitative dynamics of the local flow in a neighborhood of the codimension zero regular and singular points of the discontinuity surface.
Keywords: Regularization and discontinuous vector fields..
Mathematics Subject Classification: 58F09, 34C3.
Citation: Jaume Llibre, Marco Antonio Teixeira. Regularization of discontinuous vector fields in dimension three. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 235-241. doi: 10.3934/dcds.1997.3.235
[1]

David J. W. Simpson. On resolving singularities of piecewise-smooth discontinuous vector fields via small perturbations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3803-3830. doi: 10.3934/dcds.2014.34.3803
[2]

Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481
[3]

Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623
[4]

Luca Rondi. On the regularization of the inverse conductivity problem with discontinuous conductivities. Inverse Problems & Imaging, 2008, 2 (3) : 397-409. doi: 10.3934/ipi.2008.2.397
[5]

BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85
[6]

Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031
[7]

Alexander Krasnosel'skii, Jean Mawhin. The index at infinity for some vector fields with oscillating nonlinearities. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 165-174. doi: 10.3934/dcds.2000.6.165
[8]

Antoni Ferragut, Jaume Llibre, Adam Mahdi. Polynomial inverse integrating factors for polynomial vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 387-395. doi: 10.3934/dcds.2007.17.387
[9]

Jorge Sotomayor, Michail Zhitomirskii. On pairs of foliations defined by vector fields in the plane. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 741-749. doi: 10.3934/dcds.2000.6.741
[10]

Jaume Llibre, Claudia Valls. Centers for polynomial vector fields of arbitrary degree. Communications on Pure & Applied Analysis, 2009, 8 (2) : 725-742. doi: 10.3934/cpaa.2009.8.725
[11]

Jaume Llibre, Ricardo Miranda Martins, Marco Antonio Teixeira. On the birth of minimal sets for perturbed reversible vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 31 (3) : 763-777. doi: 10.3934/dcds.2011.31.763
[12]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247
[13]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073
[14]

Patrick Bonckaert, P. De Maesschalck. Gevrey and analytic local models for families of vector fields. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 377-400. doi: 10.3934/dcdsb.2008.10.377
[15]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020
[16]

Benedetto Piccoli, Francesco Rossi. Measure dynamics with Probability Vector Fields and sources. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6207-6230. doi: 10.3934/dcds.2019270
[17]

Patrick Foulon, Vladimir S. Matveev. Zermelo deformation of finsler metrics by killing vector fields. Electronic Research Announcements, 2018, 25: 1-7. doi: 10.3934/era.2018.25.001
[18]

Shiliang Weng, Xiang Zhang. Integrability of vector fields versus inverse Jacobian multipliers and normalizers. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6539-6555. doi: 10.3934/dcds.2016083
[19]

A.M. Krasnosel'skii, Jean Mawhin. The index at infinity of some twice degenerate compact vector fields. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 207-216. doi: 10.3934/dcds.1995.1.207
[20]

Primitivo B. Acosta-Humánez, J. Tomás Lázaro, Juan J. Morales-Ruiz, Chara Pantazi. On the integrability of polynomial vector fields in the plane by means of Picard-Vessiot theory. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1767-1800. doi: 10.3934/dcds.2015.35.1767

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (20)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences