# How is it possible to connect trajectories at a Poincare section using the "bisection" method?

When constructing a heteroclinic connection (a connection between two different periodic orbits at two different libration points), the stable and unstable manifolds of two different libration points must be connected at a defined section/plane (i.e. Poincare Section). For those who are interested, a full detailed overview is here.

However, this article mentions that "the intersection is approximated by means of some iterative process like the bisection or Newton methods". The bisection method is a numerical root-finding method for continuous functions.

I don't see how the bisection method can be used to find these intersection as sometimes the stable and unstable manifolds are discontinuous.

How can the bisection method be used to find these intersections?

When one numerically propagates the stable and unstable manifolds from discrete points on each periodic orbit, one must set a flag to stop the propagation once the Poincare section is reached. The manifolds coming from both sides do not intersect exactly at the same point on the section.

• @uhoh Thank you. That is true theoretically, they must completely be smooth and continuous. However, when you numerically propagate the stable and unstable manifolds from discrete points on each periodic orbit, you must set a flag to stop the propagation once the Poincare section is reached. The manifolds coming from both sides do not intersect exactly at the same point on the section, this is fine if you are not trying to find maneuver-free connections (the connection might right a small delta V at the intersection).
– John
Jun 15, 2020 at 8:56
• @uhoh The main reason for using the bisection method (as mentioned in couple of papers) is to find precise intersections that would construct a maneuver-free trajectory. That is my question, how is this possible when they the intersection points is not exact or have a tolerance of being at least 100 meters apart.
– John
Jun 15, 2020 at 8:58
• Oh if you are looking for something specific to numerical techniques then Scientific Computing SE might be a much better place to ask! Since cross-posting is discouraged in SE why not post this question there, and then edit this to be a little bit different so while they are related they are not duplicates? fyi I made a small edit and moved your clarification back into your question. (cf. What does “symplectic” mean in reference to numerical integrators, and does SciPy's odeint use them?)
– uhoh
Jun 15, 2020 at 9:25
• @uhoh Thank you for your suggestion and reference, seems helpful. I think this question lies somewhere between numerical techniques and constructing trajectories, as experts might state the limitations on using such techniques for constructing maneuver-free trajectories or suggest better methods.
– John
Jun 15, 2020 at 9:52
• I say this because questions about manifolds usually go unanswered here, eg Do Lissajous orbits have stable/unstable manifolds? and Did DSCOVR travel “along the stable manifold of it's future SE L1 Halo orbit” to get there? good luck!
– uhoh
Jun 15, 2020 at 9:58