Unbounded dynamics for vector fields

Consider a three-dimensional vector field $F$ which generates a finite number of fixed points - what can we say on its unbounded dynamics? In this paper we tackle this question, and prove sufficient conditions for $F$ to have fixed points with unbounded invariant manifolds. Following that, we use these results to study the dynamics of the Genesio-Tesi system, the Belousov-Zhabotinsky reaction, and the Michelson system.