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Stability of Morse—Smale Maps and Semiflows

  • Jack K. Hale
  • Luis T. Magalhães
  • Waldyr M. Oliva
Chapter
  • 477 Downloads
Part of the Applied Mathematical Sciences book series (AMS, volume 47)

Abstract

We start this Chapter by dealing with maps and Section 6.1 corresponds, precisely, to Section 10 of [87]. Later, in Section 6.2, we will present the case of semiflows. Let us consider smooth maps f: BE, B being a Banach manifold embedded in a Banach space E. The maps ƒ belong to the C r (B, E) Banach space of all E-valued C r-maps defined on B which are bounded together with their derivatives up to the order r ≥ 1. Let C r (B,B) be the subspace of C r (B,E) of all maps leaving B invariant, that is, f(B) Ì B. Denote by A(f) the set

$$A(f)\, = \,\{ \,x\, \in \,B\,:\,{\rm there}\,{\rm exists}\,{\rm a}\,{\rm sequence}\,(\,x\, = ,x_1 ,\,x_2 ,...)\, \in \,B{\rm ,}\,\mathop {\sup }\limits_j \,\left\| {x_j } \right\|\, < ,\infty \,{\rm and}\,f(x_j )\, = \,x_{j - 1} ,\,j\, = \,2,3,...\} . $$

Keywords

Periodic Orbit Periodic Point Unstable Manifold Saddle Connection Transversal Intersection 
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 New York 2002

Authors and Affiliations

  • Jack K. Hale
    • 1
  • Luis T. Magalhães
    • 2
  • Waldyr M. Oliva
    • 2
  1. 1.School of MathematicsGeorgia TechAtlantaUSA
  2. 2.Departamento de MatemáticaInstituto Superior TécnicoLisbonPortugal

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