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Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. (English) Zbl 0836.34069
From authors’ abstract: “The paper addresses, for retarded functional differential equations (FDEs), the computation of normal forms associated with the flow on a finite-dimensional invariant manifold tangent to invariant spaces for the infinitesimal generator of the linearized equation at a singularity. A phase space appropriate to the computation of these normal forms is introduced, and adequate nonresonance conditions for the computation of the normal forms are derived. As an application, the general situation of Bogdanov-Takens singularity and its versal unfolding for scalar retarded FDEs with nondegeneracy at second order is considered, both in the general case and in the case of differential- delay equations of the form $\stackrel{˙}{x}\left(t\right)=f\left(x\left(t\right),x\left(t-1\right)\right)$”.

##### MSC:
 34K05 General theory of functional-differential equations 34C20 Transformation and reduction of ODE and systems, normal forms 34C05 Location of integral curves, singular points, limit cycles (ODE) 34C23 Bifurcation (ODE)