1. CMB Online first
||On the limit cycles of linear differential systems with homogeneous nonlinearities|
We consider the class of polynomial differential systems of the
$\dot x= \lambda x-y+P_n(x,y)$, $\dot y=x+\lambda y+ Q_n(x,y),$ where
$P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. For
class of differential systems we summarize the known results
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
2. CMB 2007 (vol 50 pp. 377)
||Global Injectivity of $C^1$ Maps of the Real Plane, Inseparable Leaves and the Palais--Smale 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
half-Reeb component and the other on the Palais--Smale 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 Palais--Smale condition implies the
nonexistence of inseparable leaves, but the converse is not true.
Finally, we show that the Palais--Smale condition it is not a
necessary condition for the global injectivity of the map $X$.
3. CMB 1997 (vol 40 pp. 448)
||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
Categories:54H20, 54C60, 34C35