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 & Continuous Dynamical Systems - A, 2013, 33 (6) : 2241-2251. doi: 10.3934/dcds.2013.33.2241
References:
[1]

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

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M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", 2, 2 (1985).  doi: 10.1007/978-1-4612-4574-2.  Google Scholar

[11]

M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,", Academic Press, (1974).   Google Scholar

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

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

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show all references

References:
[1]

B. Alarcón, Rotation numbers for planar attractors of equivariant homeomorphisms,, Preprint CMUP 2012-23 and , (): 2012.   Google Scholar

[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.  doi: 10.5427/jsing.2012.6a.  Google Scholar

[3]

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

[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.  doi: 10.1080/10236190701698155.  Google Scholar

[5]

N. P. Bhatia and G. P. Szegö, "Stability Theory of Dynamical Systems,", Springer-Verlag, (2002).   Google Scholar

[6]

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

[7]

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

[8]

A. van den Essen, Conjectures and problems surrounding the Jacobian conjecture,, in, 429 ().   Google Scholar

[9]

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

[10]

M. Golubitsky, I. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", 2, 2 (1985).  doi: 10.1007/978-1-4612-4574-2.  Google Scholar

[11]

M. W. Hirsch and S. Smale, "Differential Equations, Dynamical Systems, and Linear Algebra,", Academic Press, (1974).   Google Scholar

[12]

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

[13]

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

[14]

R. Ortega, Topology of the plane and periodic differential equations,, 2008. Available from: , ().   Google Scholar

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