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I've been looking at systems which are predictable in some senses and chaotic in others. For example, consider a double pendulum that considers the movements within the pendulum body itself. The outward double pendulum is a known chaotic system, but some aspects, such as the distance between two points on the arm itself, is quite clearly predictable. They may move near or apart due to the strains from the movement, but they aren't chaotic.

Another case I'm looking at is similar to a skateboarder in a half pipe, where they just don't have enough energy to get back up to the top edge of the half pipe. This has to fail at the topological mixing criteria, as there are no regions which evolve to reach the top edge. But aside from that, the motion may be chaotic.

It seems to me that these are natural enough cases that there would be some standard ways to tease these systems apart to be analyzed by normal methods, but I can't find any. What are the standard approaches mathematicians use to refine the chaotic aspects of a system away from the predictable aspect. Or, alternatively, what approaches are used to handle both simultaneously.

Cort Ammon
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    Legrangian mechanics is one such tool. Considering the double pendulum the possible positions of the pendulum bob is limited and can be describe by the angles the arms take with respect to vertical. Paramertize the system by these constraints. Then consider that energy must be conserved, and this will limit the dymamics of the system. – user317176 Jun 14 '19 at 00:58

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TL;DR: What you describe is normal and there is nothing has to be dealt with.

Your systems are not so special. All chaotic systems have a bounded state space and thus you can always “predict” that their state will never lie outside those bounds. You also usually can make some statements about the oscillation frequencies and similar general properties of the system. More generally speaking, chaos only means that you cannot predict the state of the system, but you can still predict and characterise its dynamics.

In addition, a chaotic system can have conserved properties such as energy in the (frictionless) double pendulum or, if you so wish, the distance between the endpoints of one leg if you make the simplifying assumption that it is fixed, but include both separately into your model. Again, this is not very rare or surprising. It is also not exclusive to chaotic systems. The tools to account with this (coming mostly from theoretical mechanics) apply here as well. In particular, this is nothing that needs to be particularly “dealt with” (except maybe by using a symplectic integrator when simulating an energy-conserving system).

This has to fail at the topological mixing criteria, as there are no regions which evolve to reach the top edge.

The criterion of topological mixing only applies to the system’s phase space, not to points outside of it.

[points on the arm] may move near or apart due to the strains from the movement, but they aren't chaotic.

How do you arrive at this conclusion? If you consider an arm to be elastic, the strain on it is driven by the pendulum’s movement, which are chaotic. Hence it must be chaotic itself (though on a low magnitude).

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