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Some applications of the topological degree to stability theory

  • Chapter
Topological Methods in Differential Equations and Inclusions

Part of the book series: NATO ASI Series ((ASIC,volume 472))

Abstract

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.

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Author information

Authors and Affiliations

  1. Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, C/ Severo Ochoa s/n, E-18071, Granada, Spain

    Rafael Ortega

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  1. Rafael Ortega
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Editors and Affiliations

  1. Département de mathématiques et de statistique, Université de Montréal, Montréal, Québec, Canada

    Andrzej Granas , Marlène Frigon  & Gert Sabidussi ,  & 

  2. Faculty of Mathematics and Informatics, Nicholas Copernicus University, Toruń, Poland

    Andrzej Granas

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© 1995 Springer Science+Business Media Dordrecht

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Ortega, R. (1995). Some applications of the topological degree to stability theory. In: Granas, A., Frigon, M., Sabidussi, G. (eds) Topological Methods in Differential Equations and Inclusions. NATO ASI Series, vol 472. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0339-8_8

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  • DOI: https://doi.org/10.1007/978-94-011-0339-8_8

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4150-8

  • Online ISBN: 978-94-011-0339-8

  • eBook Packages: Springer Book Archive

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