Symmetry in Chaos

I'm not an expert, but think about linear algebra.

you can write each dimension value in terms of time t(a) -> t(b)

x = sin(t) y = 2cos(t)

or whatever. if there is a symmetry, one of those equations should cancel out.

another way to look at it is, if you can always get x, and know y "for free", or the other way around, then x and y aren't really independent. there's some other thing, like sqrt(t) that's generating both values.

To add some spice to your observation. I read somewhere that "This statement is false" can act as a oscillator, a "clock" going tick-tock, at the deepest level of reality.

Could you point to the chapter or page?

Also check out Jim "Chaos" Crutchfield's work, and the mesmerizing video feedback he made with his analog video processing computer that he built at the University of California, Santa Cruz for his Ph.D. in physics in 1984.

Jim Crutchfield

https://en.wikipedia.org/wiki/James_P._Crutchfield

https://csc.ucdavis.edu/~chaos/index.html

Space-Time Dynamics in Video Feedback

https://www.youtube.com/watch?v=B4Kn3djJMCE

A film by Jim Crutchfield, Entropy Productions, Santa Cruz (1984). Original U-matic video transferred to digital video. 16 minutes.

See:

https://csc.ucdavis.edu/~chaos/chaos/films.htm

Citation: J. P. Crutchfield, "Space-Time Dynamics in Video Feedback". Physica 10D (1984) 229-245.

https://csc.ucdavis.edu/~chaos/chaos/pubs/stdvf-title.html

Chaotic Attractors of Driven Oscillators

https://www.youtube.com/watch?v=Sq8Vu40Bw1g

A film by Jim Crutchfield, Entropy Productions, Santa Cruz (1982). Original 16mm transferred to U-matic video and then to digital video. 13 minutes.

In 2022, Crutchfield and his graduate student Kyle Ray described a way to bring the heat production of conventional circuits below the theoretical limit of Landauer's principle by encoding information not as pulses of charge but in the momentum of moving particles.

https://www.scientificamerican.com/article/lsquo-momentum-co...

While a graduate student, Crutchfield and students from the University of California, Santa Cruz (including Doyne Farmer) built a series of computers that were capable of calculating the motion of a moving roulette ball, predicting which numbers could be excluded from the outcome:

The Eudaemonic Pie / Newton's Casino: The Bizarre True Story of How a Band of Physicists and Computer Wizards Took on Las Vegas

https://archive.org/details/eudaemonicpie00bass_0/mode/2up

I have a hard time pointing you somewhere, I think you should start with the first chapter and see from there, it's a good primer.