[HN Gopher] Visualizations of Random Attractors Found Using Lyap...
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Visualizations of Random Attractors Found Using Lyapunov Exponents
Author : cs702
Score : 98 points
Date : 2025-09-30 15:50 UTC (7 hours ago)
(HTM) web link (paulbourke.net)
(TXT) w3m dump (paulbourke.net)
| zparky wrote:
| Similar, post on the Henon Attractor 4h ago:
| https://news.ycombinator.com/item?id=45424223
| cs702 wrote:
| Also, from that page:
| https://towardsdatascience.com/attractors-in-neural-network-...
| dtj1123 wrote:
| In the event that the first post led to this one, I'd be
| curious to know what the intermediate internet rabbithole
| consisted of.
| AfterHIA wrote:
| These visualizations are beautiful. I'm a musician at heart so I
| really geek out about bifurcation maps. You get to see the
| exquisite relationship between chaos and form. It's like nature
| and math producing visual jazz. Thanks for a kick ass addition
| cs702!
| elcritch wrote:
| This is how I envision LLMs working to some extent. As in that
| the "logic paths" follow something like this where the markov-
| chain-esque probabilities jump around the vector space. It
| reminds me that to get the answer I want that I need to setup the
| prompt to get me near the right "attractor logic" pathway. Once
| in a close enough ballpark then they'll bounce to the right path.
|
| As a counter, I found that if you add an incorrect statement or
| fact that lies completely outside the realm of the logic-
| attractor for a given topic that the output is severally
| degraded. Well more like a statement or fact that's "orthogonal"
| to the logic-attractor for a topic. Very much as if it's
| struggling to stay on the logic-attractor path but the outlier
| fact causes it to stray.
|
| Sometimes less is more.
| cs702 wrote:
| Interesting. Nothing prohibits us from thinking of pretrained
| LLMs as dynamical systems that take a token state and compute
| an updated token state: _x_{n+1} = LLM(x_n)_ , starting from an
| initial token state _x_0_. Surely we can compute trajectories
| (without sampling, for determinism) and study whether LLMs
| exhibit chaotic behavior. I don 't think I've seen research
| along those lines before. Has anyone here?
| elcritch wrote:
| Looks like @cs702 [1] posted a related link where a NN
| follows an attractor pattern!
|
| I've only skimmed it but it very much looks like what I've
| been imagining. It'd be cool to see more research into this
| area.
|
| 1: https://news.ycombinator.com/item?id=45427778 2:
| https://towardsdatascience.com/attractors-in-neural-
| network-...
| cs702 wrote:
| That's for a small and shallow neural network.
|
| I was wondering about LLMs specifically.
| elcritch wrote:
| Well me too, but it shows that there is some basis for
| the thinking. It's sort of surprising there's not more
| exploration into the area.
| cantor_S_drug wrote:
| https://paulbourke.net/fractals/lyapunov/
|
| > It may diverge to infinity, for the range (+- 2) used here for
| each parameter this is the most likely event. These are also easy
| to detect and discard, indeed they need to be in order to avoid
| numerical errors.
|
| https://superliminal.com/fractals/bbrot/
|
| The above image shows the overall entire Buddhabrot object. To
| produce the image only requires some very simple modifications to
| the traditional mandelbrot rendering technique: Instead of
| selecting initial points on the real-complex plane one for each
| pixel, initial points are selected randomly from the image region
| or larger as needed. Then, each initial point is iterated using
| the standard mandelbrot function in order to first test whether
| it escapes from the region near the origin or not. Only those
| that do escape are then re-iterated in a second, pass. (The ones
| that don't escape - I.E. which are believed to be within the
| Mandelbrot Set - are ignored). During re-iteration, I increment a
| counter for each pixel that it lands on before eventually
| exiting. Every so often, the current array of "hit counts" is
| output as a grayscale image. Eventually, successive images barely
| differ from each other, ultimately converging on the one above.
|
| Is it possible to use the Buddhabrot technique on the lyapunov
| fractals ?
| fractal4d wrote:
| Seems to me that the images on Bourke's site _are_ produced
| using the general "Buddhabrot" technique (splatting points onto
| an image). Although each image appears to only represent a
| single orbit sequence and the reject condition is inverted so
| that only stable orbits are shown.
|
| I've personally found the technique very versatile and have had
| a lot of fun playing around with it and exploring different
| variations. Was excited enough about the whole thing that I
| created a website for sharing some of my explorations:
| https://www.fractal4d.net/ (shameless self-advertisement)
|
| With the exception of some Mandelbrot-style images all the rest
| are produced by splatting complex-valued orbit points onto an
| image in one way or another.
| esafak wrote:
| 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-...
| MountDoom wrote:
| 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.
| sxzygz wrote:
| 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.
| MountDoom wrote:
| 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.
| westurner wrote:
| Chaos theory > Applications:
| https://en.wikipedia.org/wiki/Chaos_theory#Applications
|
| People use chaos theory to make predictions about attractor
| systems that have lower error than other models.
| cs702 wrote:
| Well, engineers building physical systems like airplanes and
| rockets use Lyapunov exponents to _avoid_ chaotic behavior. No
| one sane wants airplanes or rockets that exhibit chaotic
| aerodynamics!
|
| 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.
| DavidSJ wrote:
| To add to this, a moderate amount of turbulence (a type of
| chaotic fluid flow) in engines and wing surfaces is sometimes
| deliberately engineered to improve combustion efficiency and
| lift, and also chaotic flow can induce better mixing in heat
| exchangers and microfluidics systems.
| poslathian wrote:
| Absolutely!
|
| 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
| throwaway173738 wrote:
| Chaos is an important part of Control Systems theory from what
| I understand.
| jheitmann wrote:
| There's a book covering this and more from 1993 called "Strange
| Attractors: Creating Patterns in Chaos" by Julian C. Sprott
| that's freely available here:
| https://sprott.physics.wisc.edu/SA.HTM
|
| It's fun (errr... for me at least) to translate the ancient basic
| code into a modern implementation and play around.
|
| The article mentions that it's interesting how the 2d functions
| can look 3d. That's definitely true. But, there's also no reason
| why you can't just add on however many dimensions you want and
| get real many-dimensioned structures with which you can noodle
| around with visualizations and animations.
| throwaway173738 wrote:
| As an undergraduate I worked with some other Physics students
| to construct an analog circuit using op amps that modeled one
| of Sprott's equations and we confirmed experimentally that the
| system exhibited chaotic behavior. We also used a
| transconductance amplifier as a control parameter and swept
| through the different states (chaotic, period windows) of the
| circuit. We did not go as far as comparing the experimental and
| predicted period windows while I was there but it was an
| interesting project for us. At one point I turned up an article
| in Physica D describing how to calculate the first Lyapunov
| exponent using small data sets which we used to compute whether
| we were in a period window or not.
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