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...
Elmentve itt :
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| Dokumentumtípus: | Folyóirat |
| Megjelent: |
2016
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| Sorozat: | 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 |
| Tartalmi kivonat: | 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. |
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| Terjedelem/Fizikai jellemzők: | 29 |
| ISSN: | 1417-3875 |