June  2013, 33(6): 2241-2251. doi: 10.3934/dcds.2013.33.2241

Global dynamics for symmetric planar maps

1. 

Department of Mathematics, University of Oviedo, Calvo Sotelo s/n, 33007 Oviedo, Spain

2. 

Centro de Matemática and Faculdade de Economia, Universidade do Porto, Rua Dr. Roberto Frias, 4200-464 Porto, Portugal

3. 

Centro de Matemática, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

Received  February 2012 Revised  October 2012 Published  December 2012

We consider sufficient conditions to determine the global dynamics for equivariant maps of the plane with a unique fixed point which is also hyperbolic. When the map is equivariant under the action of a compact Lie group, it is possible to describe the local dynamics. In particular, if the group contains a reflection, there is a line invariant by the map. This allows us to use results based on the theory of free homeomorphisms to describe the global dynamical behaviour. We briefly discuss the case when reflections are absent, for which global dynamics may not follow from local dynamics near the unique fixed point.
Citation: Begoña Alarcón, Sofia B. S. D. Castro, Isabel S. Labouriau. Global dynamics for symmetric planar maps. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2241-2251. doi: 10.3934/dcds.2013.33.2241
References:
[1]

B. Alarcón, Rotation numbers for planar attractors of equivariant homeomorphisms, Preprint CMUP 2012-23 and arXiv:1206.6066, to appear in Topol. Methods Nonlinear Anal.

[2]

B. Alarcón, S. B. S. D Castro and I. Labouriau, A local but not global attractor for a $\mathbbZ_n$-symmetric map, Journal of Singularities, 6 (2012), 1-14. doi: 10.5427/jsing.2012.6a.

[3]

B. Alarcón, V. Guíñez and C. Gutierrez, Planar Embeddings with a globally attracting fixed point, Nonlinear Anal., 69 (2008), 140-150. doi: 10.1016/j.na.2007.05.005.

[4]

B. Alarcón, C. Gutierrez and J. Martínez-Alfaro, Planar maps whose second iterate has a unique fixed point, J.Difference Equ. Appl., 14 (2008), 421-428. doi: 10.1080/10236190701698155.

[5]

N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Springer-Verlag, New York, 2002.

[6]

M. Brown, Homeomorphisms of two-dimensional manifolds, Houston J. Math, 11 (1985), 455-469.

[7]

A. Cima, A. Gasull and F. Mañosas, The Discrete Markus-Yamabe Problem, Nonlinear Anal., 35 (1999), 343-354. doi: 10.1016/S0362-546X(97)00715-3.

[8]

A. van den Essen, Conjectures and problems surrounding the Jacobian conjecture, in "Recent Results on the Global Asymptotic Stability Jacobian Conjecture" (ed. M. Sabatini), Matematica 429, Dipartimento di Matematica, Universitá degli Studi di Trento.

[9]

J. Franks, A new proof of the Brouwer plane translation theorem, Ergod. Th. & Dynam. Sys., 12 (1992), 217-226. doi: 10.1017/S0143385700006702.

[10]

M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," 2, Applied Mathematical Sciences 69, Springer Verlag, 1985. doi: 10.1007/978-1-4612-4574-2.

[11]

M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, San Diego, 1974.

[12]

F. Le Roux, "Homéomorphismes de Surfaces: Théorèmes de la Fleur de Leau-Fatou et de la Variété Stable," Astérisque 292, 2004.

[13]

P. Murthy, Periodic solutions of two-dimensional forced systems: The Masera Theorem and its extension, J.Dyn and Diff Equations, 10 (1998), 275-302. doi: 10.1023/A:1022618000699.

[14]

R. Ortega, Topology of the plane and periodic differential equations, 2008. Available from: http://www.ugr.es/~ecuadif/fuentenueva.htm#Publicaciones

show all references

References:
[1]

B. Alarcón, Rotation numbers for planar attractors of equivariant homeomorphisms, Preprint CMUP 2012-23 and arXiv:1206.6066, to appear in Topol. Methods Nonlinear Anal.

[2]

B. Alarcón, S. B. S. D Castro and I. Labouriau, A local but not global attractor for a $\mathbbZ_n$-symmetric map, Journal of Singularities, 6 (2012), 1-14. doi: 10.5427/jsing.2012.6a.

[3]

B. Alarcón, V. Guíñez and C. Gutierrez, Planar Embeddings with a globally attracting fixed point, Nonlinear Anal., 69 (2008), 140-150. doi: 10.1016/j.na.2007.05.005.

[4]

B. Alarcón, C. Gutierrez and J. Martínez-Alfaro, Planar maps whose second iterate has a unique fixed point, J.Difference Equ. Appl., 14 (2008), 421-428. doi: 10.1080/10236190701698155.

[5]

N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems," Springer-Verlag, New York, 2002.

[6]

M. Brown, Homeomorphisms of two-dimensional manifolds, Houston J. Math, 11 (1985), 455-469.

[7]

A. Cima, A. Gasull and F. Mañosas, The Discrete Markus-Yamabe Problem, Nonlinear Anal., 35 (1999), 343-354. doi: 10.1016/S0362-546X(97)00715-3.

[8]

A. van den Essen, Conjectures and problems surrounding the Jacobian conjecture, in "Recent Results on the Global Asymptotic Stability Jacobian Conjecture" (ed. M. Sabatini), Matematica 429, Dipartimento di Matematica, Universitá degli Studi di Trento.

[9]

J. Franks, A new proof of the Brouwer plane translation theorem, Ergod. Th. & Dynam. Sys., 12 (1992), 217-226. doi: 10.1017/S0143385700006702.

[10]

M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," 2, Applied Mathematical Sciences 69, Springer Verlag, 1985. doi: 10.1007/978-1-4612-4574-2.

[11]

M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra," Academic Press, San Diego, 1974.

[12]

F. Le Roux, "Homéomorphismes de Surfaces: Théorèmes de la Fleur de Leau-Fatou et de la Variété Stable," Astérisque 292, 2004.

[13]

P. Murthy, Periodic solutions of two-dimensional forced systems: The Masera Theorem and its extension, J.Dyn and Diff Equations, 10 (1998), 275-302. doi: 10.1023/A:1022618000699.

[14]

R. Ortega, Topology of the plane and periodic differential equations, 2008. Available from: http://www.ugr.es/~ecuadif/fuentenueva.htm#Publicaciones

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