High School Non-linear Behavior, Chaos to Order

[Secan]

I often hear about non-linear or chaos having order, and have difficulty grasping the concept.

For example, you have weather disorder, how can it bring order? or you have Covid chaos, how can it bring order? Can you give some examples of the concept?

What is fact, myth, and misconception about it?

 

[sophiecentaur]

The sort of 'order' that comes out of chaos can best be seen in the patterns that come out of some mathematical examples. The well known Mandelbrot sets have plenty of identifiable patterns in them but none of the parts of the plot are exactly the same, as they would in the case of a classical wave pattern. That link quotes a number of naturally occurring fractal patterns which have an underlying form of order but not a rigid pattern.

The same sort of thing happens with the weather. Sometimes (not always) there are identifiable patterns in the weather and under those conditions, the 'butterfly' effect can operate. But, of course, we can't actually find that particular butterfly with which we could imagine we might control the weather. Move the butterfly a cm to the left and the result would be entirely different.

 

[Andy Resnick]

I often hear about non-linear or chaos having order, and have difficulty grasping the concept.

For example, you have weather disorder, how can it bring order? or you have Covid chaos, how can it bring order? Can you give some examples of the concept?

What is fact, myth, and misconception about it?

I'm not exactly sure what you are trying to better understand, but one form of emergent order associated with weather is the generation of large-scale coherent spatial structures (hurricanes are a good example) that arise and dissipate from underlying turbulent flow:

https://www.sciencedirect.com/science/article/abs/pii/0376042188900012

There are also related model systems, for the example the analysis of 'stadium billiards' showing emergence of stable orbits:

https://www.pks.mpg.de/~yast/Articles/T_97.pdf

 

[hutchphd]

There are also spectacular results from "cellular automata" such as John Conway's game of life. Simple rules and an astonishing panoply of stable and metastable results.

 

[anorlunda]

The so-called strange attractor is often used to illustrate the order of chaos. The graphic below shows very chaotic behavior. Do you see any order there?

 

[sophiecentaur]

Do you see any order there?

I think that was meant as a rhetorical question but there is a definite, recognisable pattern. It may be a 'bit less fuzzy' than a sphere or a cube but there is order (at least according to some definition).

 

[osilmag]

There is a good video about lorenz equations out there.

 

[Secan]

Can this concept work for the brain? What do you think?

https://www.frontiersin.org/articles/10.3389/fnmol.2017.00366/full

"
If it turned out that quantum effects cannot be observed in living systems at the macroscopic level, would that mean that living systems can be fully described by classical physics? Or is there another plausible way in which small-scale quantum effects – there is evidence for their occurrence [see (1)] – might influence large-scale neuronal activity and behavior? Yes, there is. The common view that minuscule fluctuations, including quantum events, cancel out in larger systems need not be true in highly non-linear systems like our brain. The nervous system can be seen as a nested hierarchy of non-linear complex networks of molecules, cells, microcircuits, and brain regions. In iterative hierarchies with non-linear dynamics (at the edge of chaos), small (even infinitesimal) fluctuations are not averaged out, but can be amplified. Quantum fluctuations on the lowest level of scale may influence the initial state of the next level of scale, while the higher levels shape the boundary conditions of the lower ones. This hierarchy of nested networks with many feedback loops exploits rather than cancels out the quantum effects as proposed by Jeffrey Satinover:

“[Q]uantum dynamics alters the final outcomes of computation at all levels – not by producing classically impossible solutions but by having a profound effect on which of many possible solutions are actually selected” (Satinover, 2001).

In his essay on free will and neuroscience, Haim Sompolinsky has also mentioned this possibility:

“Chaos within the brain may amplify enormously the small quantum fluctuations … to a degree that will affect the timing of spikes in neurons” (Sompolinsky, 2005).

Similarly, even Christof Koch, one of the major critics of quantum brain ideas, had to admit:

“What cannot be ruled out is that tiny quantum fluctuations deep in the brain are amplified by deterministic chaos and will ultimately lead to behavioral choices” (Koch, 2009)."