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1. CJM 2002 (vol 54 pp. 1187)

Cobo, Milton; Gutierrez, Carlos; Llibre, Jaume
On the Injectivity of $C^1$ Maps of the Real Plane
Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of (complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists $\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

Categories:34D05, 54H20, 58F10, 58F21

2. CJM 1997 (vol 49 pp. 212)

Coll, B.; Gasull, A.; Prohens, R.
Differential equations defined by the sum of two quasi-homogeneous vector fields
In this paper we prove, that under certain hypotheses, the planar differential equation: $\dot x=X_1(x,y)+X_2(x,y)$, $\dot y=Y_1(x,y)+Y_2(x,y)$, where $(X_i,Y_i)$, $i=1$, $2$, are quasi-homogeneous vector fields, has at most two limit cycles. The main tools used in the proof are the generalized polar coordinates, introduced by Lyapunov to study the stability of degenerate critical points, and the analysis of the derivatives of the Poincar\'e return map. Our results generalize those obtained for polynomial systems with homogeneous non-linearities.

Categories:34C05, 58F21

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