Weird Manifolds

So this week, I've been using the new software (which is working well now) to calculate the invariant manifolds of the map I've been researching. The best way to think of manifolds are the shapes/curves that the system points hop around on as they iterate under the mapping. There has to be saddle points in the system in order that the stable and unstable manifolds can cross and then be calculated. The calculation to find the manifolds requires tracking backwards in time the path of the saddle point...which shouldn't be possible in a noninvertible system. This is where the weird stuff comes in especially in the case for my map.

For some points, there are no points to track back to or four and for all others there are two possible previous values. Currently the software can calculate two possible values and then for each carries on the path back taking either the 'positive' or 'negative' - whichever is the most likely to be the previous value. It does factor in the no pre-images case but not yet the four possible values. That is part of the work I still have to do this year.

It helps to see some images so here are some I calculated just this week!!

The red line is the unstable manifold and the blue is the stable. The point where they cross is a saddle point in the system and the other cross is the repelling point. I've only shown the positive manifolds (one side of the possible previous values) but the negative manifolds are just the reflection in the x-axis.