Is anyone doing anything besides visualizations with this chaos stuff? I liked the article linked below depicting the state space of artificial neurons: https://towardsdatascience.com/attractors-in-neural-network-...
Has progress stalled in this area? I don't know, but surely there are people working on it. In fact I recently saw an interesting post on HN about a new technique that among other things enables faster estimation of Lyapunov exponents: https://news.ycombinator.com/item?id=45374706 (search for "Lyapunov" on the github page).
Just because we haven't seen much progress, doesn't mean we won't see more. Progress never happens on a predictable schedule.
The code and text are at https://gitlab.com/fraserphysics/hmmds. From a Nix command line, "make book" builds the book in about 10 hours.
I'd be grateful for any feedback on the book or the software.
People use chaos theory to make predictions about attractor systems that have lower error than other models.
These techniques are the key unlocks to robustifying AI and creating certifiable trust in their behavior.
Starting with pre-deep neural network era stuff like LQR-RRT trees, to the hot topic today of contraction theory, and control barrier certificates in autonomous vehicles
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.
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.
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.
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.