I did not actually try more than two iterations. I will shortly. I can’t see any reason why it won’t work but then that’s where most of the really great inventions come from, so I’ll give it a whirl.
In the mean time, and without knowing exactly what it’s supposed to look like, may I suggest approaching it from another direction?
What if you used just one arrow and placed it at its orbit distance from the logo (along one of the cardinal directions will make it easier later on). Then make a few nulls that sit at the center of the logo. Nest the nulls and then make the arrow a child of the inner most null. So Grandparent, parent, child. This way you have full control over the spin and position of the arrow without the dreaded gimbal-lock (or Heisenberg uncertainty principle for that matter). Use the Grandparent null to control say, the y spin, and only the y spin, use the parent null to control the overall z spin and the use the arrow itself to control its distance from the logo as it orbits, and also the angle of its travel direction. This way you can animate the arrow’s orbit at any point along the timeline so that it can change directions, get further out, even make the orbit become elliptical by changing the timing of the distance vs the grandparent y spin. Does that help?
Make sure to use “object” coordinates and not “world” so that all rotations and distances are in reference to the parent of the object you’re dealing with. With this system, for example, the value of the distance the arrow is from the parent null will stay the same unless you change it, even when it’s orbiting. But if you look at the arrow’s world coordinates, they change wildly as the arrow orbits – too hard to track and get a handle on this way, so keep the coordinates in object mode.
One last thing – this system won’t auto rotate the arrow so that it always points in the direction of travel. You can make an expresso that does this of course, but it’s probably easier just to animate it when needed as the key frames for this will line up exactly with the keyframes of the other rotation controlling nulls.
I’ll let you know what I find with the multiple splines being married together.