This is the 3rd in a series that started with the gravitational potential, and then the harmonic. While those are the two biggies, what we are looking at next behaves more like something you would see in real life, and doesn’t have solutions that can be written down with simple formulas.
For this entry in our series, I reached into my hat and pulled out the lorentz distribution from basic physics. It it finite in all ranges from the very small, all the way out to infinity, and dare I say beyond.
These images range from a stream of slow particles which converge directly to the center of the force field, through a range of velocities, until the force is just a blip to be zoomed over, producing just a slight deflection.
Some features to note. The intermediate images have more features and details than either the harmonic or gravitational wells. That is directly related to the cut offs in the force. Only the harmonic and gravitational potentials have elliptical orbits, all other potentials have more complicated and complex shapes.
Eventually, however the streams get fast enough that there are no major deflections and no bound orbits. That just shows up as a narrow beam of deflected particles.