For the orientation of the plane, you could use latitude and longitude of the normal vector, with a third parameter to describe the angle along the circular path. In order to get cartesian coordinates you will need to figure out the transformation (x, y, z) = f(latitude, longitude, rotation). There are no differential equations involved, just a lot of sin() and cos().
I was asked to make an animation using python, and I was thinking to animate an object making a circular path in uniform motion, but it would be really great if the plane of the circular motion also changes constantly over time (like those oversimplified atom animations), so I tried to figure some parametric equation using spherical coordinates, but I landed on differential equations something that I was supposed to learn next year. Does anyone know how to parameterize this kind of motion?
Modeling a circular trajectory with the plane of the circle changing over time