1. CMB 2015 (vol 58 pp. 818)
 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 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 

3. 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 
