# Chaotic Orbits

1. Jun 21, 2010

### Eynstone

Consider a dynamic system with a periodic trajectory. Given an arbitrary duration T of time,
does there exist a chaotic trajectory of a similar system which approximates the closed orbit
for the duration T with a given accuracy?
Chaotic orbits which I've seen so far appear to be almost periodic at times but eventually stray off. I wonder if this is a general phenomenon.

2. Jun 25, 2010

### Filip Larsen

If you have a system that exhibits chaos this system will have a region of phase space in which periodic orbits are dense, meaning that for any periodic orbit you can find another one arbitrarily close. A chaotic trajectory in such a region will indeed often look similar to a periodic orbit without actually being periodic and off the top of my head I do believe that for any such periodic orbit you can find an arbitrarily close chaotic trajectory (perhaps someone else can confirm this?).

However, note that since dense periodic orbits is a necessary but not a sufficient condition for chaos the reverse is not true, that is, a system is not necessarily chaotic just because it has dense periodic orbits. If you even more have a system with only single periodic orbit in a region (that is, periodic orbits are not dense in that region), then you can conclude that is not chaotic. I say this because I am not sure if you think of an isolated periodic orbit or not.

3. Jun 27, 2010

### Eynstone

I've some good reasons to believe this. Periodic & 'straggling' geodesics being close together is a common phenomenon. The paths of most conservative systems can be modelled as geodesics on surfaces.

4. Jun 27, 2010

### Filip Larsen

It's not clear to me where you want to go with this and if you have a question in there somewhere. If you want to pursue the matter you can perhaps describe your problem in more detail; a concrete example is usually always a good starting point.

Your original post contains two questions. The first seems to have the answer "no" under the assumption you are referring to a single isolated periodic orbit and the answer "maybe" if you are referring to dense periodic orbits. The second question can be answered with a "yes", since chaotic orbits over time by definition (i.e. sensitivity on initial conditions) will separate from any other arbitrarily close orbit.

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