Not really. Fractals and chaos theory were a bit like blockchain in that it was a "new kind of science" and it was supposed to explain everything, and you could buy pop-science books talking about the implications.
And then it sort of fizzled out, because while it's interesting and gives us a bit of additional philosophical insights into certain problems, it doesn't do anything especially useful. You can use it to draw cool space-filling shapes.
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.
I think that's a bit of a scope creep. The study of dynamical systems is obviously important and is sometimes rolled into chaos theory, but it predates it - and tellingly, it almost never concerns itself with chaotic behavior, because you can't do a whole lot with that.
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".
You can do a lot with chaos. One of the things it lets you do is find an unforced trajectory from the vicinity of any state to the vicinity of any other (accessible) state. Sensitivity to initial conditions means sensitivity to perturbations, which also means sensitivity to small control inputs, and this can be leveraged to your advantage.
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.
Fair enough. I basically had to make the not-very-compelling case that controlling non-linear systems to avoid chaos is... and application of chaos? Lol, you got me.
I don’t think you’re remotely correct, but I also don’t know how to dispute your ignorance in any useful way.
To @esafak I suggest following @westurner’s post.
I like the concept of Stable Manifolds. Classifying types of them is interesting. Group symmetries on the phase space are interesting. Explaining this and more is not work I’m prepared to do here. Use Wikipedia, ask ChatGPT, enrol in a course on Chaos and Fractal Dynamics, etc.
I am quite familiar with this space and I will reassert that its by far most significant application is making pretty pictures.
The Wikipedia list you're indirectly referencing is basically a fantasy wishlist of the areas where we expected the chaos theory to revolutionize things, with little to show for it. "Chaos theory cryptography", come on.
And then it sort of fizzled out, because while it's interesting and gives us a bit of additional philosophical insights into certain problems, it doesn't do anything especially useful. You can use it to draw cool space-filling shapes.