American Institute of Mathematical Sciences

Advanced
July  2005, 12(4): 657-674. doi: 10.3934/dcds.2005.12.657

Divergent diagrams of folds and simultaneous conjugacy of involutions

Solange Mancini 1, , Miriam Manoel 2, and  Marco Antonio Teixeira 3,

1. 

Departamento de Matemática, IGCE, Universidade Estadual Paulista, 13500-230 Caixa Postal 178, Rio Claro, SP, Brazil
2. 

Departamento de Matemática, ICMC, Universidade de São Paulo, 13560-970 Caixa Postal 668, São Carlos, SP, Brazil
3. 

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

Received  November 2003 Revised  September 2004 Published  January 2005

In this work we show that the smooth classification of divergent diagrams of folds $(f_1, \ldots, f_s) : (\mathbb R^n,0) \to (\mathbb R^n \times \cdots \times \mathbb R^n,0)$ can be reduced to the classification of the $s$-tuples $(\varphi_1, \ldots, \varphi_s)$ of associated involutions. We apply the result to obtain normal forms when $s \leq n$ and $\{\varphi_1, \ldots, \varphi_s\}$ is a transversal set of linear involutions. A complete description is given when $s=2$ and $n\geq 2$. We also present a brief discussion on applications of our results to the study of discontinuous vector fields and discrete reversible dynamical systems.
Keywords: singularities, involution, discontinuous vector fields, reversible diffeomorphisms., Divergent diagram of folds, normal form.
Mathematics Subject Classification: 34G10, 47D06, 58F11, 92D2.
Citation: Solange Mancini, Miriam Manoel, Marco Antonio Teixeira. Divergent diagrams of folds and simultaneous conjugacy of involutions. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 657-674. doi: 10.3934/dcds.2005.12.657
[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]

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
[3]

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
[4]

Ricardo Miranda Martins, Marco Antonio Teixeira. On the similarity of Hamiltonian and reversible vector fields in 4D. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1257-1266. doi: 10.3934/cpaa.2011.10.1257
[5]

Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667
[6]

Tomas Johnson, Warwick Tucker. Automated computation of robust normal forms of planar analytic vector fields. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 769-782. doi: 10.3934/dcdsb.2009.12.769
[7]

Gianluca Crippa, Milton C. Lopes Filho, Evelyne Miot, Helena J. Nussenzveig Lopes. Flows of vector fields with point singularities and the vortex-wave system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2405-2417. doi: 10.3934/dcds.2016.36.2405
[8]

C. Alonso-González, M. I. Camacho, F. Cano. Topological invariants for singularities of real vector fields in dimension three. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 823-847. doi: 10.3934/dcds.2008.20.823
[9]

Tiago de Carvalho, Bruno Freitas. Birth of an arbitrary number of T-singularities in 3D piecewise smooth vector fields. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4851-4861. doi: 10.3934/dcdsb.2019034
[10]

Vivi Rottschäfer. Multi-bump patterns by a normal form approach. Discrete & Continuous Dynamical Systems - B, 2001, 1 (3) : 363-386. doi: 10.3934/dcdsb.2001.1.363
[11]

Todor Mitev, Georgi Popov. Gevrey normal form and effective stability of Lagrangian tori. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 643-666. doi: 10.3934/dcdss.2010.3.643
[12]

Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109
[13]

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
[14]

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
[15]

Ricardo Miranda Martins. Formal equivalence between normal forms of reversible and hamiltonian dynamical systems. Communications on Pure & Applied Analysis, 2014, 13 (2) : 703-713. doi: 10.3934/cpaa.2014.13.703
[16]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362
[17]

Letizia Stefanelli, Ugo Locatelli. Kolmogorov's normal form for equations of motion with dissipative effects. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2561-2593. doi: 10.3934/dcdsb.2012.17.2561
[18]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[19]

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
[20]

Robert Roussarie. A topological study of planar vector field singularities. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5217-5245. doi: 10.3934/dcds.2020226

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (37)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences