A generically immersed loop in the plane can have double points but does not have triple points and self-tangencies.  However, in a generic homotopy triple points and self-tangencies can occur.  Arnold's $J^+$-invariant does not change under triple intersections and inverse self-tangencies but is sensible to direct self-tangencies.  This is of interest for force laws which give rise to second order ODE's for which direct self-tangencies are forbidden, but triple intersections can occur as well as inverse self-tangencies in the case the force depends not just on position but as well on velocity.  Periodic orbits are not always immersed but can have cusps and collisions.  With Kai Cieliebak and Otto van Koert we constructed invariants for families of periodic orbits based on Arnold's $J^+$-invariant which I plan to discuss in my talk.