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Realization of Vector Fields and Normal Forms

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

Abstract

The meaning of realization comes from the following observation. Since the flow for a RFDE evolves in an infinite dimensional space, for any given integers N and n, it is perhaps to be expected that, for any system of ordinary differential equations of dimension N, there is an RFDE of dimension n such that the flow for the RFDE can be mapped onto the flow for the ODE. Of course, if nN, this is the case. If n < N, we will observe below that there are flows defined by an N dimensional ODE which cannot be realized by an RFDE in ℝ n . We verify this observation by considering the restrictions imposed on the ODE which determines the flow on a center manifold.

Keywords

Normal Form Hopf Bifurcation Invariant Manifold Center Manifold Infinitesimal Generator 
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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