Let $X:\,{{\mathbb{R}}^{2}}\to \,{{\mathbb{R}}^{2}}$ be a ${{C}^{1}}$ map. Denote by $\text{Spec}(X)$ the set of (complex) eigenvalues of $\text{D}{{\text{X}}_{p}}$ when $p$ varies in ${{\mathbb{R}}^{2}}$. If there exists $\in \,>\,0$ such that $\text{Spec(}X)\,\bigcap \,(-\in ,\,\in )\,=\,\varnothing $, then $X$ is injective. Some applications of this result to the real Keller Jacobian conjecture are discussed.

In this paper we prove, that under certain hypotheses, the planar differential equation: ˙x = X1(x, y) + X2(x, y), ˙y = Y1(x, y) + Y2(x, y), where (Xi, Yi), 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.

In this article we study the simultaneous generation of limit cycles out of singular points and infinity for the family of cubic planar systems

With a suitable choice of parameters, the origin and four other singularities are foci and infinity is a periodic orbit. We prove that it is possible to obtain the following configuration of limit cycles: two small amplitude limit cycles out of the origin, a small amplitude limit cycle out of each of the other four foci, and a large amplitude limit cycle out of infinity. We also obtain other configurations with fewer limit cycles.