# What is the relation between universality and chaotic behavior in Elementary Cellular Automata?

So i know that rule 110 from Elementary Cellular Automata is universal but i am wondering if we had other chaotic behaviors let's say there exists a system that shows nearly infinite chaotic behaviors {a structure at a time} similar to ECA structures, how can we prove or disprove universality from that system ?

Suppose we have a system with one specific rule to execute and the input can be defined by different number of cells {i.e input = 5 cells} these 5 cells for example can show (2000) chaotic behavior... 4 cells can show nearly (1800), for example to get a structure from the (1800) structures, i can change the 4 cells by changing the rule set of these 4 cells just like in ECA => 'two white cells will result in a black cell', i can define a similar rule set on the 4 cells, 5 cells or more generally on x cells where x < infinite.

That system with same input {4 cells} with different rule set can simulate most of the structures in Elementary Cellular Automata like rule 30, 90 & 60 and others, what can be proved about such system and how universality is related to it ?

• You prove (or disprove) universality by proving that the thing can (or cannot) simulate some other universal system. – David Richerby Oct 13 '15 at 23:01
• Can you define "chaotic behaviors", or what it means for a system to "show nearly infinite chaotic behaviors", in a precise or rigorous way? If not then you certainly can't prove a result of the form you seem to hope exists. Personally, I doubt that any such result exists. There are chaotic systems that are very simple and I doubt are universal. The question seems to be based on a hope that somehow there is some relationship between chaos vs universality, but I'm skeptical about whether that hope is actually valid. – D.W. Oct 13 '15 at 23:09

For instance, here's an example to challenge your intuition that the two concepts are somehow related. Wikipedia lists the dynamic system given by the map $x \mapsto 4x (1-x)$ and $y \mapsto x + y \bmod 1$ as an example of a chaotic system. However I doubt that this system is universal in any meaningful sense.