American Institute of Mathematical Sciences

Advanced
July  2000, 6(3): 519-536. doi: 10.3934/dcds.2000.6.519

On pairs of differential $1$-forms in the plane

R.D.S. Oliveira 1, and  F. Tari 1,

1. 

Universidade de São Paulo, Instituto de Ciências Matemáticas e de Computação, Caixa Postal 668, CEP 13560-970, São Carlos (SP), Brazil, Brazil

Received  May 1999 Revised  March 2000 Published  April 2000

We classify pairs of germs of differential $1$-forms $(\alpha, beta)$ in the plane, where $\alpha$, $beta$ are either regular or have a singularity of type saddle/node/focus. The main tools used here are singularity theory and the method of polar blowing up. We also present a desingularization theorem for pairs of germs of differential $1$-forms in the plane.
Keywords: desingularization., Differential 1-forms, pairs of foliations.
Mathematics Subject Classification: 34C20, 34A34, 58C2.
Citation: R.D.S. Oliveira, F. Tari. On pairs of differential $1$-forms in the plane. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 519-536. doi: 10.3934/dcds.2000.6.519
[1]

Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159
[2]

Yuri B. Suris. Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms. Journal of Geometric Mechanics, 2013, 5 (3) : 365-379. doi: 10.3934/jgm.2013.5.365
[3]

Jorge Sotomayor, Michail Zhitomirskii. On pairs of foliations defined by vector fields in the plane. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 741-749. doi: 10.3934/dcds.2000.6.741
[4]

Erica Clay, Boris Hasselblatt, Enrique Pujals. Desingularization of surface maps. Electronic Research Announcements, 2017, 24: 1-9. doi: 10.3934/era.2017.24.001
[5]

Xinsheng Wang, Lin Wang, Yujun Zhu. Formula of entropy along unstable foliations for $C^1$ diffeomorphisms with dominated splitting. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2125-2140. doi: 10.3934/dcds.2018087
[6]

Carlos Gutierrez, Víctor Guíñez, Alvaro Castañeda. Quartic differential forms and transversal nets with singularities. Discrete & Continuous Dynamical Systems, 2010, 26 (1) : 225-249. doi: 10.3934/dcds.2010.26.225
[7]

Holger Heumann, Ralf Hiptmair, Cecilia Pagliantini. Stabilized Galerkin for transient advection of differential forms. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 185-214. doi: 10.3934/dcdss.2016.9.185
[8]

Olivier Hénot. On polynomial forms of nonlinear functional differential equations. Journal of Computational Dynamics, 2021, 8 (3) : 307-323. doi: 10.3934/jcd.2021013
[9]

Jun Shen, Kening Lu, Bixiang Wang. Invariant manifolds and foliations for random differential equations driven by colored noise. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6201-6246. doi: 10.3934/dcds.2020276
[10]

Zhongkai Guo. Invariant foliations for stochastic partial differential equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5203-5219. doi: 10.3934/dcds.2015.35.5203
[11]

Weigu Li, Jaume Llibre, Hao Wu. Polynomial and linearized normal forms for almost periodic differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 345-360. doi: 10.3934/dcds.2016.36.345
[12]

Holger Heumann, Ralf Hiptmair. Eulerian and semi-Lagrangian methods for convection-diffusion for differential forms. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1471-1495. doi: 10.3934/dcds.2011.29.1471
[13]

Dorina Mitrea and Marius Mitrea. Boundary integral methods for harmonic differential forms in Lipschitz domains. Electronic Research Announcements, 1996, 2: 92-97.

[14]

Dorina Mitrea, Irina Mitrea, Marius Mitrea, Lixin Yan. Coercive energy estimates for differential forms in semi-convex domains. Communications on Pure & Applied Analysis, 2010, 9 (4) : 987-1010. doi: 10.3934/cpaa.2010.9.987
[15]

Yvette Kosmann-Schwarzbach. Dirac pairs. Journal of Geometric Mechanics, 2012, 4 (2) : 165-180. doi: 10.3934/jgm.2012.4.165
[16]

Teresa Faria. Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II. Discrete & Continuous Dynamical Systems, 2001, 7 (1) : 155-176. doi: 10.3934/dcds.2001.7.155
[17]

Víctor Guíñez, Eduardo Sáez. Versal Unfoldings for rank--2 singularities of positive quadratic differential forms: The remaining case. Discrete & Continuous Dynamical Systems, 2005, 12 (5) : 887-904. doi: 10.3934/dcds.2005.12.887
[18]

Hsin-Yi Liu, Hsing Paul Luh. Kronecker product-forms of steady-state probabilities with $C_k$/$C_m$/$1$ by matrix polynomial approaches. Numerical Algebra, Control & Optimization, 2011, 1 (4) : 691-711. doi: 10.3934/naco.2011.1.691
[19]

Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93
[20]

Radu Saghin. Note on homology of expanding foliations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 349-360. doi: 10.3934/dcdss.2009.2.349

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences