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

Walther Hans-Otto: Local invariant manifolds for delay differential equations with state space in C1((−∞, 0], R n). (2016)

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Abstract

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.

Item Type: Journal
Other title: Honoring the career of Tibor Krisztin on the occasion of his sixtieth birthday
Publication full: Electronic journal of qualitative theory of differential equations : special edition
Date: 2016
Volume: 2
Number: 85
ISSN: 1417-3875
Number of Pages: 29
Language: English
DOI: 10.14232/ejqtde.2016.1.85
Uncontrolled Keywords: Differenciálegyenlet - késleltetett
Additional Information: Bibliogr.: p. 28-29. ; összefoglalás angol nyelven
Date Deposited: 2021. Nov. 11. 15:18
Last Modified: 2021. Nov. 12. 09:42
URI: http://acta.bibl.u-szeged.hu/id/eprint/73752

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