1. CMB Online first
 Llibre, Jaume; Zhang, Xiang

On the limit cycles of linear differential systems with homogeneous nonlinearities
We consider the class of polynomial differential systems of the
form
$\dot x= \lambda xy+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where
$P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For
this
class of differential systems we summarize the known results
for the
existence of limit cycles, and we provide new results for their
nonexistence and existence.
Keywords:polynomial differential system, limit cycles, differential equations on the cylinder Categories:34C35, 34D30 

2. CMB 2011 (vol 56 pp. 39)
 Ben Amara, Jamel

Comparison Theorem for Conjugate Points of a Fourthorder Linear Differential Equation
In 1961, J. Barrett showed that if the first conjugate point
$\eta_1(a)$ exists for the differential equation $(r(x)y'')''=
p(x)y,$ where $r(x)\gt 0$ and $p(x)\gt 0$, then so does the first
systemsconjugate point $\widehat\eta_1(a)$. The aim of this note is to
extend this result to the general equation with middle term
$(q(x)y')'$ without further restriction on $q(x)$, other than
continuity.
Keywords:fourthorder linear differential equation, conjugate points, systemconjugate points, subwronskians Categories:47E05, 34B05, 34C10 

3. CMB 2011 (vol 56 pp. 366)
 Kyritsi, Sophia Th.; Papageorgiou, Nikolaos S.

Multiple Solutions for Nonlinear Periodic Problems
We consider a nonlinear periodic problem driven by a
nonlinear nonhomogeneous differential operator and a
CarathÃ©odory reaction term $f(t,x)$ that exhibits a
$(p1)$superlinear growth in $x \in \mathbb{R}$
near $\pm\infty$ and near zero.
A special case of the differential operator is the scalar
$p$Laplacian. Using a combination of variational methods based on
the critical point theory with Morse theory (critical groups), we
show that the problem has three nontrivial solutions, two of which
have constant sign (one positive, the other negative).
Keywords:$C$condition, mountain pass theorem, critical groups, strong deformation retract, contractible space, homotopy invariance Categories:34B15, 34B18, 34C25, 58E05 

4. CMB 2011 (vol 55 pp. 3)
5. CMB 2009 (vol 53 pp. 193)
6. CMB 2009 (vol 52 pp. 315)
 Yi, Taishan; Zou, Xingfu

Generic QuasiConvergence for Essentially Strongly OrderPreserving Semiflows
By employing the limit set
dichotomy for essentially strongly orderpreserving semiflows and
the assumption that limit sets have infima and suprema in the
state space, we prove a generic quasiconvergence principle
implying the existence of an open and dense set of stable
quasiconvergent points. We also apply this generic quasiconvergence principle
to a model for biochemical feedback in protein
synthesis and obtain some results about the model which are of theoretical
and realistic significance.
Keywords:Essentially strongly orderpreserving semiflow, compactness, quasiconvergence Categories:34C12, 34K25 

7. 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 

8. 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 

9. 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 

10. 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 
