Geodesics dynamics in the Linet-Tian spacetime with Λ > 0

We analyse the geodesics' dynamics in cylindrically symmetric vacuum spacetimes with Lambda > 0 and compare it to the Lambda = 0 and Lambda < 0 cases. When Lambda > 0 there are two singularities in the metric which brings new qualitative features to the dynamics. We find that Lambda = 0 planar timelike confined geodesics are unstable against the introduction of a sufficiently large Lambda, in the sense that the bounded orbits become unbounded. In turn, any non-planar radially bounded geodesics are stable for any positive Lambda. We construct global non-singular static vacuum spacetimes in cylindrical symmetry with Lambda > 0 by matching the Linet-Tian metric with two appropriate sources.