I don't see how better understanding non-linear systems and global dynamics can be not be considered useful. For starters, better control of nonlinear systems/keeping them from turning them chaotic is incredibly useful. So many hard problems can be approximately reduced to "keep this non-linear system stable." Staying in the "edge of chaos" regime has proven to be an optimal choice for a plethora of problems.
So it's sort of like saying that the physics of black holes are very useful to us day-to-day because we want to make sure we don't fall into any black holes.
I'm not saying that chaos theory isn't interesting. It's just that it's pretty hard to find any concrete application of it, beyond hand-wavy stuff like "oh, it somehow helped us understand weather".
Multibody orbits are one such chaotic system, which means you can take advantage of that chaos to redirect your space probe from one orbit to another using virtually zero fuel, as NASA did with its ISEE-3 spacecraft.