It's what typically occurs in all deterministically chaotic systems.

Stam Nicolis , thank you for your answer :)...I see what you mean...but

could you provide an eample of a deterministic spatiotemporal many body system that allows for the definition of an order parameter such that, when driven to (spatiotemporal) chaos, shows critical behavior? and percolation, power laws, etc etc those typicl features of a phase transition as well?

I mean, Ising model would not apply because is fed by noise, spatiotemporal Kuramoto neither because is fed by diversity.....

I would like to compare such system with mine....

The logistic map shows such behavior, when all the periods 2^oo become unstable. This isn't a spatially extended system., of course.

thank you for taking your time for this conversation...that's very kind of you... I've checked your profile and I could see that you're very competent and skilled! ...yes I'm actually looking for spatiotemporal systems because it's where pattern formation takes places :)

have a nice day!

One example of such systems are the so-called coupled-map lattices.

great! I check it out!

You're welcome! You may want to check the paper ``Arnol'd cat map lattices"...

Sure!

if you've time please have a look at

The emergence of dynamic networks from many coupled polar oscillators. A model for Artificial Life (published in Theory in Biosciences - July 2023)

is that strange dynamics familar to you??

Dear Allesandro,

I am not sure (mathematically, with respect to the common accepted definition of your question - disclosure: I am an experienced IT expert and not a mathematician) but iSpace theory - or better said its geometry based on the rarely known concept of the so called „changed distance definition“ - able to derive 50+ constants of nature in 2 fully independent unit systems (iSpace-SI and iSpace-IQ) plus accepted Planck unit system might very well be considered as another „deterministic phase transiton other than cellular automata's“ by compatibility to the implicit both numerical and integer geometric exactness of cellular automata.

Article New novel physical constants metric and fine structure const...

Preprint Hubble constant H0 is derived from Newtonian gravitational c...

Preprint iSpace - Quantization of Time in iSpace-IQ Unit-System by 1/...

Conference Paper iSpace - Deriving G from α, e, R∞, μ0 and Quantum of Gravita...

Conference Paper iSpace - Exact Symbolic Equations for Important Physical Con...

Being an invited member of 2012 3 week Wolfram Research NKS (A new kind of science at the time, today titled Wolfram Summer School), I consider myself by no means an expert on cellular automata, but one might say i’d able to claim reasonable insight into this topic.

https://www.researchgate.net/profile/Christian-Wolf-20

Hello Christian,

thank for your reply!!

I'll check it out :D

There is the known fact that partial differential equations can be casted into cellular automaton implementing discretized scheme of PDEs. There ar e two articles of Toffoli and Vichniac that discuss this relationship in quite nice forms. Both are in the same journal issue.

https://www.sciencedirect.com/science/article/abs/pii/0167278984902549

Additionally, I discuss those issues in the future version of the introduction of the review paper on Emergent information processing:

https://www.researchgate.net/publication/373044641_EMERGENT_INFORMATION_PROCESSING_OBSERVATIONS_EXPERIMENTS_AND_FUTURE_DIRECTIONS

The revised version will be uploaded soon in coming weeks.

I recommend to study logistic equations and dynamical systems. The work of Poincare on three body problem, Lorenz model of weather, etc. It might be interesting to read Hermann Haken's books on synergetics.

Phase transitions are associated with the term period doubling in discrete dynamical systems.

Jiří Kroc thank you!!! :D

After re-reading your question, try to read Hermann Haken's books on Synergetics and Advanced Synergetics. It will give you the solid overview of the area you are working in. This might help too.