Alarcón, B.; Castro, S. B. S. D.; Labouriau, I. S. Global saddles for planar maps. (English) Zbl 1394.37049 J. Dyn. Differ. Equations 30, No. 2, 601-612 (2018). Summary: We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of \(D_2\)-symmetric maps, for which we obtain a similar result for \(C^1\) homeomorphisms. Some applications to differential equations are also given. MSC: 37C75 Stability theory for smooth dynamical systems 37C80 Symmetries, equivariant dynamical systems (MSC2010) 37C05 Dynamical systems involving smooth mappings and diffeomorphisms 37C20 Generic properties, structural stability of dynamical systems 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems Keywords:planar map; symmetry; local and global dynamics; saddle PDFBibTeX XMLCite \textit{B. Alarcón} et al., J. Dyn. Differ. Equations 30, No. 2, 601--612 (2018; Zbl 1394.37049) Full Text: DOI arXiv References: [1] Alarcón, B, Rotation numbers for planar attractors of equivariant homeomorphisms, Topol. Methods Nonlinear Anal., 42, 327-343, (2013) · Zbl 1301.37014 [2] Alarcón, B; Castro, SBSD; Labouriau, I, A local but not global attractor for a \(\mathbb{Z}_n\)-symmetric map, J. Singul., 6, 1-14, (2012) · Zbl 1292.37002 [3] Alarcón, B; Castro, SBSD; Labouriau, I, Global dynamics for symmetric planar maps, Discret. Contin. Dyn. Syst., A, 37, 2241-2251, (2013) · Zbl 1317.37036 [4] Alarcón, B; Gutierrez, C; Martínez-Alfaro, J, Planar maps whose second iterate has a unique fixed point, J. Differ. Equ. Appl., 14, 421-428, (2008) · Zbl 1149.26024 · doi:10.1080/10236190701698155 [5] Brown, M, Homeomorphisms of two-dimensional manifolds, Houst. J. Math, 11, 455-469, (1985) · Zbl 0605.57005 [6] Campos, J; Torres, PJ, On the structure of the set of bounded solutions on a periodic Liénard equation, Proc. Am. Math. Soc., 127, 1453-1462, (1999) · Zbl 0920.34045 · doi:10.1090/S0002-9939-99-05046-7 [7] van den Essen, A.: Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, 190 Birkhäuser Verlag, (2000) · Zbl 0962.14037 [8] Golubitsky, M., Stewart, I., Schaeffer, D.G.: Singularities and groups in bifurcation theory. Applied Mathematical Sciences, vol. 2. Springer, New York (1985) · Zbl 0691.58003 · doi:10.1007/978-1-4612-5034-0 [9] Gutierrez, C, A solution to the bidimensional global asymptotic conjecture, Ann. Inst. Poincaré Anal. Non Linéaire, 12, 627-671, (1995) · Zbl 0837.34057 · doi:10.1016/S0294-1449(16)30147-0 [10] Gutierrez, C; Martínez-Alfaro, J; Venato-Santos, J, Plane foliations with a saddle singularity, Topol. Appl., 159, 484-491, (2012) · Zbl 1238.34092 · doi:10.1016/j.topol.2011.09.023 [11] Hirsch, M, Fixed-point indices, homoclinic contacts, and dynamics of injective planar maps, Mich. Math. J., 47, 101-108, (2000) · Zbl 0978.37035 · doi:10.1307/mmj/1030374670 [12] Murthy, P, Periodic solutions of two-dimensional forced systems: the masera theorem and its extension, J. Dyn. Differ. Equ., 10, 275-302, (1998) · Zbl 0914.34042 · doi:10.1023/A:1022618000699 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.