26. CMB 2007 (vol 50 pp. 377)
 Gutierrez, C.; Jarque, X.; Llibre, J.; Teixeira, M. A.

Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the PalaisSmale Condition
We study two sufficient conditions that imply global injectivity
for a $C^1$ map $X\colon \R^2\to \R^2$ such that its Jacobian at any
point of $\R^2$ is not zero. One is based on the notion of
halfReeb component and the other on the PalaisSmale condition.
We improve the first condition using the notion of inseparable
leaves. We provide a new proof of the sufficiency of the second
condition. We prove that both conditions are not equivalent, more
precisely we show that the PalaisSmale condition implies the
nonexistence of inseparable leaves, but the converse is not true.
Finally, we show that the PalaisSmale condition it is not a
necessary condition for the global injectivity of the map $X$.
Categories:34C35, 34H05 

27. CMB 2006 (vol 49 pp. 170)
 Atkins, Richard

The Geometry of $ d^{2}y^{1}/dt^{2} = f(y, \dot{y},t) \; \text{and} \; d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$, and Euclidean Spaces
This paper investigates the relationship between a system of
differential equations and the underlying geometry associated with
it. The geometry of a surface determines shortest paths, or
geodesics connecting nearby points, which are defined as the
solutions to a pair of secondorder differential equations: the
EulerLagrange equations of the metric. We ask when the converse
holds, that is, when solutions to a system of differential
equations reveals an underlying geometry. Specifically, when may
the solutions to a given pair of second order ordinary
differential equations $d^{2}y^{1}/dt^{2} = f(y,\dot{y},t)$ and
$d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$ be reparameterized by
$t\rightarrow T(y,t)$ so as to give locally the geodesics of a
Euclidean space? Our approach is based upon Cartan's method of
equivalence. In the second part of the paper, the equivalence
problem is solved for a generic pair of second order ordinary
differential equations of the above form revealing the existence
of 24 invariant functions.
Category:34A26 

28. CMB 2005 (vol 48 pp. 405)
29. CMB 2002 (vol 45 pp. 355)
 Cresson, Jacky

Obstruction Ã la linÃ©arisation 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 \'EcalleVallet.
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
\'EcalleVallet. Then, we extend Schuman's result about non
existence of isochronous homogenous Hamiltonian systems.
Keywords:linÃ©arisationproblÃ¨me du centrehamiltoniendarbouxchamps polynomiaux Categories:34D10, 34D30 

30. 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 

31. 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 

32. CMB 1998 (vol 41 pp. 214)
33. CMB 1998 (vol 41 pp. 23)
 Clemence, Dominic P.

Subordinacy analysis and absolutely continuous spectra for SturmLiouville equations with two singular endpoints
The GilbertPearson characterization of the spectrum is established
for a generalized SturmLiouville 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 

34. CMB 1997 (vol 40 pp. 416)
35. 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 

36. CMB 1997 (vol 40 pp. 276)
 Chouikha, Raouf

Fonctions elliptiques et Ã©quations diffÃ©rentielles 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 
