Abstract
In this paper, we prove that a stable isolated fixed point of an orientation preserving local homeomorphism onR 2 has fixed point index 1. We also give a number of applications to differential equations. In particular, we deduce that a number of existence methods for producing periodic solutions of differential equations in the plane always produce unstable solutions.
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Amann, H. (1982). A note on degree theory for gradient mappings.Proc. Am. Math. Soc. 85, 591–595.
Brown, M. (1984). A new proof of Brouwer's lemma on translation arcs.Houston Math. J. 10, 35–41.
Brown, M. (1985). Homeomorphisms of two-dimensional manifolds.Houston Math. J. 11, 455–469.
Erle, D. (1993). Stable equilibria and vector field index.Topol. Appl. 49, 231–235.
Krasnoselskii, M. A. (1968).Translation Along Trajectories of Differential Equations, Am. Math. Soc., Providence, RI.
Krasnoselskii, M. A., and Zabreiko, P. P. (1984).Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin.
Krasnoselskii, M. A., Perov, A. I., Povolotskii, A. I., and Zabreiko, P. P. (1966).Plane Vector Fields, Academic Press, New York.
Levi-Civita, T. (1901). Sopra alcuni criteri di instabilita.Ann. Mat. V, 221–307.
Massera, J. L. (1950). The existence of periodic solutions of systems of differential equations.Duke Math. J 17, 457–475.
Mawhin, J. (1985).Points Fixes, Points Critiques et Problèmes aux Limites, Les Presses de l'Université de Montréal, Montréal.
Ortega, R. (1990). Topological degree and stability of periodic solutions for certain differential equations.J. London Math. Soc. 42, 505–516.
Pliss, V. A. (1966).Nonlocal Problems of the Theory of Oscillations, Academic Press, New York.
Siegel, C. L., and Moser, J. K. (1971).Lectures on Celestial Mechanics, Springer-Verlag, Berlin.
Spanier, E. H. (1989).Algebraic Topology, Springer-Verlag, Berlin.
Thews, K. (1989). Der Abbildungsgrad von Vektorfeldern zu stabilen Ruhelagen.Arch. Math. 52, 71–74.
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Dancer, E.N., Ortega, R. The index of lyapunov stable fixed points in two dimensions. J Dyn Diff Equat 6, 631–637 (1994). https://doi.org/10.1007/BF02218851
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DOI: https://doi.org/10.1007/BF02218851