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26. 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

27. CMB 2001 (vol 44 pp. 323)

Schuman, Bertrand
Une classe d'hamiltoniens polynomiaux isochrones
Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan $\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de $H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\`ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\`ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the plane $\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian $H_0$ considered in [L], [P], [S].

Keywords:Hamiltonian system, normal forms, resonance, linearization
Categories:34C20, 58F05, 58F22, 58F30

28. CMB 1998 (vol 41 pp. 207)

Philos, Ch. G.; Sficas, Y. G.
An oscillation criterion for first order linear delay differential equations
A new oscillation criterion is given for the delay differential equation $x'(t)+p(t)x \left(t-\tau(t)\right)=0$, where $p$, $\tau \in \C \left([0,\infty),[0,\infty)\right)$ and the function $T$ defined by $T(t)=t-\tau(t)$, $t\ge 0$ is increasing and such that $\lim_{t\to\infty}T(t)=\infty$. This criterion concerns the case where $\liminf_{t\to\infty} \int_{T(t)}^{t}p(s)\,ds\le \frac{1}{e}$.

Keywords:Delay differential equation, oscillation
Category:34K15

29. CMB 1998 (vol 41 pp. 214)

Shackell, John
On a problem of Rubel concerning the set of functions satisfying all the algebraic differential equations satisfied by a given function
For two functions $f$ and $g$, define $g\ll f$ to mean that $g$ satisfies every algebraic differential equation over the constants satisfied by $f$. The order $\ll$ was introduced in one of a set of problems on algebraic differential equations given by the late Lee Rubel. Here we characterise the set of $g$ such that $g\ll f$, when $f$ is a given Liouvillian function.

Categories:34A34, 12H05

30. CMB 1998 (vol 41 pp. 23)

Clemence, Dominic P.
Subordinacy analysis and absolutely continuous spectra for Sturm-Liouville equations with two singular endpoints
The Gilbert-Pearson characterization of the spectrum is established for a generalized Sturm-Liouville equation with two singular endpoints. It is also shown that strong absolute continuity for the one singular endpoint problem guarantees absolute continuity for the two singular endpoint problem. As a consequence, we obtain the result that strong nonsubordinacy, at one singular endpoint, of a particular solution guarantees the nonexistence of subordinate solutions at both singular endpoints.

Categories:34L05, 34B20, 34B24

31. CMB 1997 (vol 40 pp. 416)

32. CMB 1997 (vol 40 pp. 448)

Kaczynski, Tomasz; Mrozek, Marian
Stable index pairs for discrete dynamical systems
A new shorter proof of the existence of index pairs for discrete dynamical systems is given. Moreover, the index pairs defined in that proof are stable with respect to small perturbations of the generating map. The existence of stable index pairs was previously known in the case of diffeomorphisms and flows generated by smooth vector fields but it was an open question in the general discrete case.

Categories:54H20, 54C60, 34C35

33. CMB 1997 (vol 40 pp. 276)

Chouikha, Raouf
Fonctions elliptiques et équations différentielles ordinaires
In this paper, we detail some results of a previous note concerning a trigonometric expansion of the Weierstrass elliptic function $\{\wp(z);\, 2\omega, 2\omega'\}$. In particular, this implies its classical Fourier expansion. We use a direct integration method of the ODE $$(E)\left\{\matrix{{d^2u \over dt^2} = P(u, \lambda)\hfill \cr u(0) = \sigma\hfill \cr {du \over dt}(0) = \tau\hfill \cr}\right.$$ where $P(u)$ is a polynomial of degree $n = 2$ or $3$. In this case, the bifurcations of $(E)$ depend on one parameter only. Moreover, this global method seems not to apply to the cases $n > 3$.

Categories:33E05, 34A05, 33E20, 33E30, 34A20, 34C23
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