We analyse the intersection of positively and negatively sectional-hyperbolic sets for flows on compact manifolds. First we prove that such an intersection is hyperbolic if the intersecting sets are both transitive (this is false without such a hypot
hesis). Next we prove that, in general, such an intersection consists of a nonsingular hyperbolic set, finitely many singularities and regular orbits joining them. Afterward we exhibit a three-dimensional star flow with two homoclinic classes, one being positively (but not negatively) sectional-hyperbolic and the other negatively (but not positively) sectional-hyperbolic, whose intersection reduces to a single periodic orbit. This provides a counterexample to a conjecture by Shy, Zhu, Gan and Wen (cite{sgw}, cite{zgw}).
We prove that every sectional-Anosov flow of a compact 3-manifold $M$ exhibits a finite collection of hyperbolic attractors and singularities whose basins form a dense subset of $M$. Applications to the dynamics of sectional-Anosov flows on compact 3
-manifolds include a characterization of essential hyperbolicity, sensitivity to the initial conditions (improving cite{ams}) and a relationship between the topology of the ambient manifold and the denseness of the basin of the singularities.
