7

One of the generic ways to obtain chaos in phase space is when the system causes trajectories to stretch and fold.

I understand that the stretching will cause neighboring initial conditions to diverge, which is one of the conditions of chaos, but why is there need for folding?

I’ve also been told that folding into a finite domain is necessary for chaos. Why? I honestly cannot picture this in my head. Why can’t we fold and stretch into an expanding domain and still have chaos?

Wrzlprmft
  • 5,718
ddc
  • 71
  • The folding leads to a complex behaviour. If you are interested in chaotic cellular automata,google "rule 110" . It is fascinating! – Peter Jan 29 '18 at 01:29
  • If you just keep stretching without folding your space back into a finite domain, won't your trajectories just diverge to infinity? –  Jan 29 '18 at 02:57
  • @ddc your question reminded me the algorithm that architect Michael Hansmeyer uses to fold three-dimensional cubes to obtain different fractal-like patterns and construct physical columns with them. It is a three years old project but could be related with your topic: https://www.youtube.com/watch?v=pACNKOLZizE&list=WL&index=8 – iadvd Jan 29 '18 at 09:25

2 Answers2

3

The short answer is that chaos was defined to only contain recurring dynamics – which is typically formally captured in the requirement of topological mixing. If you just stretch, your dynamics cannot be recurring, otherwise the respective part of phase space would not have been stretched. If you stretch and fold in a non-finite domain, it cannot be recurring either: Whatever governs the size of your domain must be encoded in your phase space somehow and if the domain is ever growing, this property/state is not recurring.

Now, the more interesting question is why chaos was defined this way. Sensitivity to initial conditions in non-recurring dynamics is easy to produce, just consider $f(x)=3x$ or $\dot{x} = 2x$ (which both contain only stretching). This was already known and understood long before chaos theory and it is not particularly exciting if you ask me. Moreover, there is not much overlap between studying such phenomena and chaos theory. Hence, when the term chaos became established for the collection of topics now known as chaos theory, it made sense to define the term such that it captures the object of interest of chaos theory and only that. Otherwise we would be talking about finite chaos or something similar today.

Wrzlprmft
  • 5,718
1

Stretching alone isn't enough to call a system "chaotic", as you note. Well, neither is folding. It's possible to have a system that does a very regular kind of folding; think of squaring in the complex plane. One of the most studied examples of a chaotic map is $z \mapsto z^2 + c$; if $c=0$ we don't get chaos because it's just folding. But you do see points getting 'mixed up' in a way, since you can find two points in the preimage for most points in the range. You might say that stretching makes points that start close get more distant, and folding makes points that are distant get closer. It's the interaction between the two that makes things fun.

Sort of Damocles
  • 2,443
  • That's actually interesting -- why isn't the squaring map, or even its restriction to the unit circle $\theta \mapsto 2\theta\bmod2\pi$, chaotic? It has sensitive dependence on initial conditions after all: the distance between two nearby trajectories increases exponentially. –  Jan 29 '18 at 02:55
  • 2
    @Rahul: Unless I am mistaken, the restriction to the unit circle chaotic: We have the butterfly effect, topological mixing, and dense periodic orbits. It is also very similar to the bit-shift map. Without the restriction, however, the unit circle becomes a chaotic repellor with the map stretching and folding inside and just stretching outside. – Wrzlprmft Jan 29 '18 at 10:08