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Symmetry in Chaos

paulbourke.net

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Re: Symmetry in Chaos

#41

Post by abetusk »

alimw wrote: post #31
abetusk wrote: post #20

These are awesome, but it would be nice to understand, even at some heuristic level, why they're periodic with the period they do have. Alternatively, what is a change to a completely periodic orbit to chaos with the same "periodic symmetry" that gives some enlightenment of where the chaos is being inserted and why it isn't destroying the periodic orbit. Does anyone have an idea of whats' going on, or references that…

All of these are aperiodic; it says as much in the first paragraph. Are you talking about the rotational symmetry?

Yes, the high level rotational symmetry. There's some gross level structure that's preserved while the individual points are (presumably) completely aperiodic/chaotic.

What's some insight into the gross level symmetric features appearing? How do you convert something that's completely periodic to an aperiodic/chaotic system with gross level periodic structure? What are the operations that allow the gross level periodic structure to remain while making the fine grained structure aperiodic/chaotic?

Re: Symmetry in Chaos

#42

Post by passion__desire »

andybak wrote: post #33
passion__desire wrote: post #23

Earlier quoted context omitted.

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.

Ok. I need to know more.

Well actually, it was a comment on a blogpost (I don't know what it was about). It said since if our world is a simulation, then it needs a computer to run on. And every computer needs a clock. So this statement changing its truth value could act as a clock.

Re: Symmetry in Chaos

#43

Post by dmbche »

abetusk wrote: post #40
dmbche wrote: post #29

Earlier quoted context omitted.

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.

I've skimmed Gleick's book before and my feeling is that it's a high level overview of the subject without much content in either explaining the underlying math, the motivation behind it or giving some deeper insight into the subject, at least at a level that I would consider valuable. I'm probably being too pejorative, but these books (like GEB or the like) are something I consider "feel good" books that give the il…

It's been a while, but I remember many experiments being presented in Gleick's book ( The double pendulum being a chaotic system, or the paper showing how it was impossible to predict the distance between two points in a structure after steching and folding it)

The gist being that some systems are much more sensitive than others to initial variables - and these are what we call chaotic.

For the double pendulum, for example, you try to release the pendulum from the same point, with 0 force - no matter how precise you are, there will be variability in the position, air pressure, temperature, turbulence and whatnot that will induce a small variability. This is unavoidable - but in chaotic systems, the effects are impossible to predict i.e. the pendulum always swings differently.

The pretty pictures come from what are called "strange attractors" which are all dependent on the system you are studying because they flow from the systems attributes. It's not black magic - a simple example from the book is that heat distribution can be chaotic in liquid systems like coffee cups - so it's impossible to predict the precise temperature inside any cup given it's initial state, but we all know it's cold after an hour. This is not the best explanation for this concept, I believe the last third of the book is on the subject.

Hopefully this could help out.

Edit: if you were to plot position of the pendulum for many thousands of drops, some patterns will emerge - the pendulum moves in a "finite" set of positions, because it physically can't go in some positions after some others, for example. Or it will have the same period in rising and falling in all graphs from having close to the same initial energy but not orientation.

So it'll look neat - but it doesn't necessarily mean much.

Re: Symmetry in Chaos

#44

Post by abetusk »

dmbche wrote: post #43
abetusk wrote: post #40

Earlier quoted context omitted.

I've skimmed Gleick's book before and my feeling is that it's a high level overview of the subject without much content in either explaining the underlying math, the motivation behind it or giving some deeper insight into the subject, at least at a level that I would consider valuable. I'm probably being too pejorative, but these books (like GEB or the like) are something I consider "feel good" books that give the il…

It's been a while, but I remember many experiments being presented in Gleick's book ( The double pendulum being a chaotic system, or the paper showing how it was impossible to predict the distance between two points in a structure after steching and folding it) The gist being that some systems are much more sensitive than others to initial variables - and these are what we call chaotic. For the double pendulum, for e…

OK, that's fair, I was being too critical of Gleick's book.

Even so, this doesn't really get at the high level symmetry. Gleick's book might give some motivation for the chaotic points being restricted to a compact domain (space? manifold? area/volume?) but I don't see how to make the leap to the highly structured gross level symmetry in the OP.

Re: Symmetry in Chaos

#45

Post by alimw »

abetusk wrote: post #41
alimw wrote: post #31

Earlier quoted context omitted.

All of these are aperiodic; it says as much in the first paragraph. Are you talking about the rotational symmetry?

Yes, the high level rotational symmetry. There's some gross level structure that's preserved while the individual points are (presumably) completely aperiodic/chaotic. What's some insight into the gross level symmetric features appearing? How do you convert something that's completely periodic to an aperiodic/chaotic system with gross level periodic structure? What are the operations that allow the gross level period…

They iterate f where f(z) = (a₀ + a₁ z ź + a₂ Re(zⁿ) + a₃ i) z + a₄ źⁿ⁻¹. You can easily check that f(z exp(i 2 π / n)) = exp(i 2 π / n) f(z).

Re: Symmetry in Chaos

#48

Post by dmbche »

abetusk wrote: post #44
dmbche wrote: post #43

Earlier quoted context omitted.

It's been a while, but I remember many experiments being presented in Gleick's book ( The double pendulum being a chaotic system, or the paper showing how it was impossible to predict the distance between two points in a structure after steching and folding it) The gist being that some systems are much more sensitive than others to initial variables - and these are what we call chaotic. For the double pendulum, for e…

OK, that's fair, I was being too critical of Gleick's book. Even so, this doesn't really get at the high level symmetry. Gleick's book might give some motivation for the chaotic points being restricted to a compact domain (space? manifold? area/volume?) but I don't see how to make the leap to the highly structured gross level symmetry in the OP.

I think you might want to look into the Mandelbrot and Julia set, and things like Kochs Snowflake, as these are purely mathematical systems like in OP - this is one part of the book I didn't integrate as well, you might be satisfied by the book - but OP also lists some references on the symmetry of chaotic systems, maybe that's a better way forward!

Re: Symmetry in Chaos

#49

Post by abetusk »

dmbche wrote: post #48
abetusk wrote: post #44

Earlier quoted context omitted.

OK, that's fair, I was being too critical of Gleick's book. Even so, this doesn't really get at the high level symmetry. Gleick's book might give some motivation for the chaotic points being restricted to a compact domain (space? manifold? area/volume?) but I don't see how to make the leap to the highly structured gross level symmetry in the OP.

I think you might want to look into the Mandelbrot and Julia set, and things like Kochs Snowflake, as these are purely mathematical systems like in OP - this is one part of the book I didn't integrate as well, you might be satisfied by the book - but OP also lists some references on the symmetry of chaotic systems, maybe that's a better way forward!

See alimw's response [0].

[0] https://news.ycombinator.com/item?id=37114898

Re: Symmetry in Chaos

#50

Post by lanstin »

abetusk wrote: post #20

These are awesome, but it would be nice to understand, even at some heuristic level, why they're periodic with the period they do have. Alternatively, what is a change to a completely periodic orbit to chaos with the same "periodic symmetry" that gives some enlightenment of where the chaos is being inserted and why it isn't destroying the periodic orbit. Does anyone have an idea of whats' going on, or references that…

These systems do not have periodic orbits they have symmetric strange attractors.

So tracing a particle thru these systems will not result in periodicity but will trace out these symmetric structures. It is unclear from the website if these are proven symmetries or observed.

A good book that is a bit more technical than Gleick (which I found not wrong in the details it does have) is “Introduction to Applied Nonlinear Dynamical Systems and Chaos” by Stephen Wiggins. It requires basic under grad maths but is a graduate text in that very close reading is needed to follow. The first 17 chapters are intended as a semester class.

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