One of the things that confused my when I started reading about chaos is easy to
explain using what we've covered about attractors. (The image to the side was created by Jean-Francois Colonna, and is part of his slide-show here)
Here's the problem: We know that things like N-body gravitational systems are chaotic - and a common example of that is how a gravity-based orbital system that appears stable for a long time can suddenly go through a transition where one body is violently ejected, with enough velocity to permanently escape the orbital system.
But when we look at the definition of chaos, we see the requirement for dense periodic orbits. But if a body is ejected from a gravitational system, ejection of a body from a gravitational system is a demonstration of chaos, how can that system have periodic orbits?
The answer relates to something I mentioned in the last post. A system doesn't have to be
chaotic at all points in its phase space. It can be chaotic under some conditions
- that is, chaotic in some parts of the phase space. Speaking loosely, when a phase space has
chaotic regions, we tend to call it a chaotic phase space.
In the gravitational system example, you do have a region of dense periodic orbits. You can create an N-body gravitational system in which the bodies will orbit forever, never actually repeating a configuration, but also never completely breaking down. The system will never repeat. Per Ramsey theory, given any configuration in its phase space, it must eventually come arbitrarily close to repeating that configuration. But that doesn't mean that it's really repeating: it's chaotic, so even those infinitesimal differences will result in
divergence from the past - it will follow a different path forward.
An attractor of a chaotic system shows you a region of the phase space where the system behaves
chaotically. But it's not the entire phase space. If the attractor covered the entire
space, it wouldn't be particularly interesting or revealing. What makes it interesting
is that it captures a region where you get chaotic behavior. The attractor isn't the whole
story of a chaotic systems phase space - it's just one interesting region with useful analytic
So to return to the N-body gravitational problem: the phase space of an N-body
gravitational system does contain an attractor full of dense orbits. It's definitely
very sensitive to initial conditions. There are definitely phase spaces for N-body
systems that are topologically mixing. None of that
precludes the possibility that you can create N-body gravitational systems that
break up and allow escape. The escape property isn't a good example of the chaotic
nature of the system, because it encourages people to focus on the wrong properties
of the system. The system isn't chaotic because you can create gravitational
systems where a body will escape from what seemed to be a stable system. It's chaotic
because you can create systems that don't break down, which are stable,
but which are thoroughly unpredictable, and will never repeat a configuration.
The system will never repeat. Per Ramsey theory, given any configuration in its phase space, it must eventually come arbitrarily close to repeating that configuration. But that doesn't mean that it's really repeating: it's chaotic, so even those infinitesimal differences will result in divergence from the past - it will follow a different path forward.
For some reason, this made me think of Penrose tiling. You should do a post on that some time.
The subject of this post is very appropriate considering the content of the previous. :)
I only just found this blog through reddit, but good lord, both this and all your other math posts are fantastic. Thankyou.
I think the N-body system is a confusing example because it does not have an attractor even though it is chaotic. Your definition of an attractor A requires that A has a trapping neighborhood B. But since energy is conserved in the N-body problem the time-1 image of a bounded set B cannot be a proper subset of B.
Chaotic dynamical systems are very easy (to get wrong). After all, the whole subject started with a mistake by Poincare. My own knowledge is limited, so comments from experts on this subject would probably be really helpful.
To #4: Usually chaos is defined with respect to an invariant closed set. It doesn't have to be an attractor. An extreme example is a Julia set in complex dynamics, which is if anything a "strange repellor".
The co-existence of a dense orbit with dense periodic cycles isn't as mysterious as it seems. The key is to think of the shift map in symbolic dynamics, which makes all sorts of aspects of chaotic behavior seem downright obvious.