Some applications of the topological degree to stability theory
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These notes are devoted to showing that the topological degree is a useful tool in the study of the properties of stability of periodic solutions of a scalar, time-dependent differential equation of Newtons type. Two different situations are considered depending on whether the equation has damping or not. When there is linear friction the asymptotic stability of a periodic solution can be characterized in terms of degree. When there is no friction the equation has a hamiltonian structure and some connections between Lyapunov stability and degree are discussed. These results are applied in two different directions: to prove that some classical methods in the theory of existence lead to instability (minimization of the action functional, upper and lower solutions) and to study the stability of the solutions of a concrete class of equations (equations of pendulum-type).
The general results are presented in an abstract setting also applicable to other two-dimensional periodic systems.
KeywordsPeriodic Solution Asymptotic Stability Stability Theory Degree Theory Periodic Problem
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- Brown, M., A new proof of Brouwers lemma on translation arcs, Houston Math. J. 10 (1984), 455–469.Google Scholar
- Castro, A., Periodic solutions of the forced pendulum equation, in: Differential Equations (S. Ahmad, M. Keener, A. Lazer, eds.), Academic Press, New York, 1980.Google Scholar
- Dancer, E. N., Ortega, R., The index of Lyapunov stable fixed points in two dimensions, J. Dynamics Differential Equations 6 (1994).Google Scholar
- Ioos, G., Bifurcation of Maps and Applications, North-Holland, New York, 1979.Google Scholar
- Krasnoselskii, M.A., The Operator of Translation Along the Trajectories of Differential Equations, Transl. Math. Monographs 19, American Mathematical Society, Providence RI, 1968.Google Scholar
- Krasnoselskii, M., Perov, A., Poloskiy, A., Zabreiko, P., Plane Vector Fields, Academic Press, New York, 1966.Google Scholar
- Magnus, W., Winkler, S., Hills Equation, Dover, New York, 1979.Google Scholar
- Mawhin, J., Equations fonctionnelles non lineaires et solutions periodiques, in: Equadiff 70, Centre de Recherches Physiques, Marseille, 1970.Google Scholar
- Mawhin, J. Compacité, monotonie et convexité dans létude de problèmes aux lirnites semilinéaires, Séminaire dAnalyse Moderne, no 19., Univ. de Sherbrooke, 1981.Google Scholar
- Ortega, R., Stability and index of periodic solutions of an equation of Duffing type, Boll. Un. Mat. Ital. 3-B (1989), 533–546.Google Scholar
- Ortega, R., A criterion for asymptotic stability based on topological degree, in: Proc. First World Congress of Nonlinear Analysts, Tampa, 1992.Google Scholar
- Protter M., Weinberger H., Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs NJ, 1967.Google Scholar
- Simo, C, Stability of degenerate fixed points of analytic area preserving maps, Asterisque 98–99 (1982), 184–194.Google Scholar