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Some applications of the topological degree to stability theory

  • Rafael Ortega
Chapter
Part of the NATO ASI Series book 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.

Keywords

Periodic Solution Asymptotic Stability Stability Theory Degree Theory Periodic Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1995

Authors and Affiliations

  • Rafael Ortega
    • 1
  1. 1.Departamento de Matemática Aplicada, Facultad de CienciasUniversidad de GranadaGranadaSpain

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