Marrakech 2002

Normal Forms and Bifurcations for Delay Differential Equations

M.T. Faria

U. Lisbonne
Portugal

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:

Normal forms for FDEs in Rⁿ.
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.

Selected references

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