Can Chaotic Trajectories Approximating Periodic Orbits Exist Over Time?

[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.

 

[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.

 

[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 modeled as geodesics on surfaces.

 

[Filip Larsen]

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 modeled as geodesics on surfaces.

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.