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Behaviour near a hyperbolic equilibrium

  • Odo Diekmann
  • Sjoerd M. Verduyn Lunel
  • Stephan A. van Gils
  • Hanns-Otto Walther
Part of the Applied Mathematical Sciences book series (AMS, volume 110)

Abstract

In this chapter we start to investigate the behaviour of the nonlinear semiflow near an equilibrium. Throughout this chapter, and also in Chapters IX and X, we consider X over ℝ. For the linearized semiflow {T(t)}, the time asymptotic behaviour near the equilibrium u ≡ 0 is described by the decomposition of the state space according to the spectrum of the generator of the semigroup and the accompanying exponential dichotomy. In this chapter we will assume that there is no spectrum on the imaginary axis. In the case of RFDE, the spectrum in the right half-plane consists of finitely many, say k: eigenvalues (counting multiplicity) with a positive real part. From Chapter IV we recall that in this case we can decompose X as
$$X\, = {X_ - } \oplus {X_{ + \cdot }}$$

Keywords

Unstable Manifold Stable Manifold Exponential Dichotomy Bounded Linear Mapping Stable Subspace 
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-Verlag New York, Inc. 1995

Authors and Affiliations

  • Odo Diekmann
    • 1
    • 2
  • Sjoerd M. Verduyn Lunel
    • 3
  • Stephan A. van Gils
    • 4
  • Hanns-Otto Walther
    • 5
  1. 1.Centrum voor Wiskunde en InformaticaAmsterdamThe Netherlands
  2. 2.Instituut voor Theoretische BiologieRijkuniversiteit LeidenLeidenThe Netherlands
  3. 3.Faculteit der Wiskunde en InformaticaUniversiteit van AmsterdamAmsterdamThe Netherlands
  4. 4.Faculteit der Toegepaste WiskundeUniversiteit TwenteEnschedeThe Netherlands
  5. 5.Mathematisches InstitutJustus-Leibig-UniversitätGiessenGermany

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