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.