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.
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.
> 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.
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 ?
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!
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.
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-...
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.
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 ?
Similar, post on the Henon Attractor 4h ago: https://news.ycombinator.com/item?id=45424223
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!