# Counting closed orbits on a compact surface topologically equivalent to a sphere

Previously I asked a question about a surface formed by gluing together pieces of spheres, or, contracting overlayed copies of spheres, or configuring $$4$$ twice pointed spheres inside a region of volume $$1$$: How does one find the systole of the $$2$$-manifold? Although "systole" may not have been the right term to describe the blue bands, it seems like "geodesic" is the correct term.

I'm interested in the dynamics on this surface (which is a sort of smoothed stellated octahedron). Something that makes me unsure that dynamical analysis can be done on this surface, is that the surface is not everywhere smooth (it has the $$3$$ "rubber band curves"). Also if a geodesic test particle travels across one of these rubber band paths I'm not sure if one can logically figure out the trajectory of the test particle after it crosses the crevice (there's not one single derivative, depends on which direction one approaches the path from). Although the surface does not seem ideal to to analysis on I still would like to understand the dynamics on this surface if possible.

Note: "Theorem of the $$3$$ geodesics." (This theorem states that every Riemannian manifold with the topology of a sphere has at least $$3$$ closed geodesics that form simple closed curves without self intersections. The result can also be extended to quasigeodesics on a convex polyhedron.

I'm interested in the closed geodesics (like the "rubber band curves"). Are there any other closed geodesics besides these $$3$$? How can I count all of them?

I think Clairut's relation, which states that $$r(t)\cos\theta(t)=constant,$$ could help but I might need some hints.