Looping Shake

Needing some help looping the shake in Fusion.

Camera Shake, or the Shake modifier?

Shake modifier

A brute force method is to plot the curve and check if you find a value that matches and has a similar tangent. Then you could mirror it. I am not sure what shake uses but I'd say its goal was to be not repeating that much. It just needs to be consistent when the same seed and values are present.

You could also come up with your own function that has a random looking shape.
E.g. try:
sin(x)+sin(sin(x)*5)-cos(x*0.6)

on this website:
http://rechneronline.de/function-graphs/
Set the x range to -30 to 30.
You will see it repeating. You could change its amplitude and frequency (with radians) to match an area which interests you for your animation. x will be time in Fusion expressions.

But this should be repeating over and over again. As long as you stick to sine and cosine there will be a time where the shape repeats.