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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:

In summary, the topics covered in this course are the following:

- Normal forms for FDEs in R
^{n}. - Applications to the study of Hopf and Bogdanov-Takens bifurcations.
- Normal forms for FDEs in infinite dimensional spaces.
- Applications to the study of reaction-diffusion equations with delays.
- Formal adjoint theory for linear FDEs in general Banach spaces and normal forms: a general framework.

- T. Faria, Normal forms and Hopf bifurcation for partial
differential equations with delays,
*Transactions of the AMS*,**352**(2000), 2217-2238. Link to this paper on the AMS site. - T. Faria, Stability and bifurcation for a delayed
predator-prey model and the effect of diffusion,
*J. Math. Anal. Appl.***254**(2001), 433-463. Link to this paper on the JMAA site. - T. Faria, Normal forms for semilinear functional differential
equations in Banach spaces and applications, Part II,
*Disc Cont. Dyn. Systems*,**7**(2001), 155-176. - T. Faria and W. Huang, Stability of periodic solutions
arising from Hopf bifurcation for a reaction-diffusion
equation with time delay,
*Fields Inst. Commun.***31**(2002), 125-141. - T. Faria, W. Huang and J. Wu,
Smoothness of center manifolds for maps and formal adjoints
for semilinear functional
differential equations in general Banach spaces
(to appear in
*SIAM J. Math. Anal.*). - T. Faria and L.T. Magalhaes,
Normal forms for retarded functional differential
equations and applications to Bogdanov-Takens singularity,
*J. Differential Equations***122**(1995), 201-224. Link to this paper on the JDE site. - T. Faria and L.T. Magalhaes, Normal forms for retarded
functional differential equations with parameters and
applications to Hopf singularity,
*J. Differential Equations***122**(1995), 181-200. Link to this paper on the JDE site. - J. K. Hale and S. M. Verduyn Lunel, ``Introduction to Functional Differential Equations", Springer-Verlag, New-York, 1993.
- J. K. Hale and M. Weedermann, On perturbations of delay-differential equations with periodic orbits, preprint.
- X. Lin, J. W.-H. So and J. Wu, Centre manifolds for
partial differential
equations with delays,
*Proc. Roy. Soc. Edinburgh***122A**(1992), 237--254. - M. C. Memory, Stable and unstable manifolds for
partial functional differential
equations,
*Nonlinear Anal. TMA***16**(1991), 131--142. - M. Weedermann, Normal forms for neutral functional
differential equations,
*Fields Inst. Commun.***29**(2001), 361--368. - J. Wu, ``Theory and Applications of Partial Functional Differential Equations", Springer-Verlag, New York, 1996.

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