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Search: MSC category 34D10 ( Perturbations )

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1. CMB Online first

Preda, Ciprian
On the roughness of quasinilpotency property of one–parameter semigroups
Let $\mathbf{S}:=\{S(t)\}_{t\geq0}$ be a C$_0$-semigroup of quasinilpotent operators (i.e. $\sigma(S(t))=\{0\}$ for each $t\gt 0$). In the dynamical systems theory the above quasinilpotency property is equivalent to a very strong concept of stability for the solutions of autonomous systems. This concept is frequently called superstability and weakens the classical finite time extinction property (roughly speaking, disappearing solutions). We show that under some assumptions, the quasinilpotency, or equivalently, the superstability property of a C$_0$-semigroup is preserved under the perturbations of its infinitesimal generator.

Keywords:one-parameter semigroups, quasinilpotency, superstability, essential spectrum
Categories:34D05, 34D10, 34E10

2. CMB 2002 (vol 45 pp. 355)

Cresson, Jacky
Obstruction à la linéarisation des champs de vecteurs polynomiaux
On explicite une classe de champ de vecteurs polynomiaux non analytiquement lin\'earisables \`a l'aide de la correction introduite par \'Ecalle-Vallet. Notamment, on \'etend des r\'esultats de Schuman sur la trivialit\'e des hamiltoniens homog\`enes isochrones. We characterize a class of polynomial vector fields which are not analytically linearizable using the correction introduced by \'Ecalle-Vallet. Then, we extend Schuman's result about non existence of isochronous homogenous Hamiltonian systems.

Keywords:linéarisation-problème du centre-hamiltonien-darboux-champs polynomiaux
Categories:34D10, 34D30

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