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Normal forms and Hopf bifurcation for partial differential equations with delays. (English) Zbl 0955.35008

The normal forms on center manifolds (or other invariant manifolds) for Partial Functional Differential Equations (PFDEs)

$\frac{d}{dt}u\left(t\right)=d{\Delta }u\left(t\right)+L\left({u}_{t}\right)+F\left({u}_{t}\right),\phantom{\rule{1.em}{0ex}}t>0\phantom{\rule{2.em}{0ex}}\left(1\right)$

near equilibrium points were computed, and then the qualitative behavior (namely when a Hopf bifurcation accures) of solutions was studied; where $d>0$, $\text{dom}\left({\Delta }\right)\subset X$, $L:𝒞\to X$ is a bounded linear operator, and $F:𝒞\to X$ is a ${C}^{k}$ function ($k\ge 2$) such that $F\left(0\right)=0$, $DF\left(0\right)=0$; $𝒞=C\left(\left[-r,0\right];X\right)\phantom{\rule{4pt}{0ex}}\left(r>0\right)$ is the Banach space with the sup norm; $X$ is a Hilbert space of functions from $\overline{{\Omega }}$ to ${ℝ}^{m}$ with inner product $〈·,·〉$ and ${\Omega }\subset {ℝ}^{n}$ is open. It turns out that the coeffients of the normal forms are explicitly given in terms of the coeffients of the original PFDE (1). With the approach presented here, the author gave explicit normal forms (in the usual sense of ordinary differential equations) for the equation giving the flow on the center manifold, without having to compute the manifold beforehand. In the particular case of generic Hopf bifurcation near equilibria, the author showed that, under certain conditions, that normal form coincides (up to the third order terms) with the normal form of the FDE associated (in the precise and natural way) with the given PFDE. In this article, the author adopted the hypotheses and most of the notations in the article of X. Lin, J. W.-H. So and J. Wu [Proc. R. Soc. Edinb., Sect. A 122, No. 3-4, 237-254 (1992; Zbl 0801.35062)], and followed the work of himself and L. T. Magalhães [J. Differ. Equations 122, No. 2, 181-200 (1995; Zbl 0836.34068)] for autonomous retarded functional differential equations.

As an illustration of the procedure, two examples of PFDEs where a Hopf singularity occurs on the center manifold are considered.

MSC:
 35B32 Bifurcation (PDE) 35R10 Partial functional-differential equations 34K17 Transformation and reduction of functional-differential equations and systems; normal forms 34K30 Functional-differential equations in abstract spaces 37L10 Normal forms, center manifold theory, bifurcation theory
center manifold