Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II

Pages: 155 - 176, Volume 7, Issue 1, January 2001

doi:10.3934/dcds.2001.7.155       Abstract        Full Text (264.5K)       Related Articles

Teresa Faria - Departamento de Matemática, Faculdade de Ciências, and CMAF, Universidade de Lisboa, Campo Grande, 1749-016 Lisboa, Portugal (email)

Abstract: A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.

Keywords:  Normal form theory, functional differential equations.
Mathematics Subject Classification:  34K30, 34K17, 34K18, 35K57.

Received: May 2000;      Revised: August 2000;      Published: November 2000.