Marrakech 2002

Normal Forms and Bifurcations for Delay Differential Equations

M.T. Faria

U. Lisbonne

In this course, the theory of normal forms for functional differential equations (FDEs) in finite and infinite dimensional spaces is presented. In applications, normal forms are usually considered for the ordinary differential equation (ODE) associated with the restriction of the flow to center manifolds of singularities. The present theory allows the normal form for this ODE to be obtained without having to compute the center manifold beforehand. These normal forms are also applicable to determine the flow on other finite dimensional invariant manifolds, provided that certain nonresonance conditions hold. Since this normal form theory is particularly powerful in the study of bifurcation problems, special emphasis will be given to applications.
In summary, the topics covered in this course are the following:
  1. Normal forms for FDEs in Rn.
  2. Applications to the study of Hopf and Bogdanov-Takens bifurcations.
  3. Normal forms for FDEs in infinite dimensional spaces.
  4. Applications to the study of reaction-diffusion equations with delays.
  5. Formal adjoint theory for linear FDEs in general Banach spaces and normal forms: a general framework.

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Last modified: Sun Sep 08 23:56:09 Pacific Daylight Time 2002