Local invariant manifolds for delay differential equations with state space in C1((−∞, 0], R n)

Consider the delay differential equation x 0 (t) = f(xt) with the history xt : (−∞, 0] → Rn of x at ‘time’ t defined by xt(s) = x(t + s). In order not to lose any possible entire solution of any example we work in the Fréchet space C 1 ((−∞, 0], Rn with the topology of uniform convergence of maps an...

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Bibliographic Details
Main Author: Walther Hans-Otto
Format: Serial
Published: 2016
Series:Electronic journal of qualitative theory of differential equations : special edition 2 No. 85
Kulcsszavak:Differenciálegyenlet - késleltetett
doi:10.14232/ejqtde.2016.1.85

Online Access:http://acta.bibl.u-szeged.hu/73752
Description
Summary:Consider the delay differential equation x 0 (t) = f(xt) with the history xt : (−∞, 0] → Rn of x at ‘time’ t defined by xt(s) = x(t + s). In order not to lose any possible entire solution of any example we work in the Fréchet space C 1 ((−∞, 0], Rn with the topology of uniform convergence of maps and their derivatives on compact sets. A previously obtained result, designed for the application to examples with unbounded state-dependent delay, says that for maps f which are slightly better than continuously differentiable the delay differential equation defines a continuous semiflow on a continuously differentiable submanifold X ⊂ C 1 of codimension n, with all time-t-maps continuously differentiable. Here continuously differentiable for maps in Fréchet spaces is understood in the sense of Michal and Bastiani. It implies that f is of locally bounded delay in a certain sense. Using this property – and a related further mild smoothness hypothesis on f – we construct stable, unstable, and center manifolds of the semiflow at stationary points, by means of transversality and embeddings.
Physical Description:29
ISSN:1417-3875