When I was a senior in college, I went through the math of the double pendulum. The subject is mostly just ordinary differential equations although there is a cute role for a little matrix theory.

Generally it seems to me that the OP gets off into some not very well defined and not very relevant topics and, instead, for the 'chaos' he is observing there's a fairly easy explanation: The system is unstable. Or, in one step more detail, the system really is an initial value problem for an ordinary differential equation, but, going way back to Bellman's work on stability theory, it's long been well known, just from the equations, that the solution can be 'unstable', that is, small changes in the initial conditions (values) can result in large changes in the solution. And that is just from ordinary differential equations without considering quantum mechanics. And for the 'chaos' in the OP, that's about all the explanation that is needed.

Why? Because there is really no chance that the motion of the system could be periodic or even simple because it nearly never gets back accurately enough to an earlier state. So, the system often gets back to something 'close' to an earlier state, but the system is so unstable that 'close' is not close enough so that the earlier state and the present one close to that earlier state soon result in very different solutions for the future.

Something similar happens with pseudo random number generators, e.g., the usual linear congruential generators where we set

R(n + 1) = ( A * R(n) + B ) mod C

for n = 1, 2, ..., and R(1) some
positive integer. Then roughly
the R(n)/C are independent, identically
distributed, uniform on [0,1). One
of Knuth's recommendations (in one of the
volumes of TACP) was
A = 5^15, B = 1, C = 2^47. So, a
point here is, get such 'random'
numbers without considering phase
space or quantum mechanics.

In a sense, this is a very old point:
E.g., one dream before about 1900 was
that we could observe the present
state of the world and, then, use deterministic
physics to predict the future. So,
as I recall, it was E. Borel who
did a calculation and concluded that
a change of moving 1 gram of matter
1 cm, or some such, on a distant star would invalidate
predictions on earth after just milliseconds
(presumably starting after the travel
time of light from that star to the earth).

We suspect we see much the same in
weather prediction: Small changes in
initial conditions too soon make changes
large enough to switch between rain
and sunshine. The usual joke is that
a butterfly could flap its wings and
convert a clear day to a hurricane.

We anticipate that probabilistically
weather prediction is quite stable, that
is, what is stable is the conditional
probability distribution of the variables
we use to measure weather conditioned
on the present. In particular, we still
believe in the law of conservation of
energy.

Also we should notice the classic work
on ergodic theory, by Hopf, Poincare,
Birkhoff, etc.: The standard illustration is
pouring cream into coffee and stirring.
Then the theorem says that, if stir long
enough, then can make the cream separate
from the coffee back to as close as please
to the original state. Why? Because
if take the 'volume' of the possible
states at some point in time and then
let time pass, then the 'volume' of
those states is still the same. So,
as the system evolves, it is 'measure
preserving' in state space. So, if
want to apply this to a frictionless
double pendulum, then can get it to
return as close as we please to its
initial state, but between then and
now it is free to do a lot. This
stuff goes back to the first half
of the last century.

Likely there are some interesting
and important questions in chaos theory,
but what the OP is saying about the
double pendulum seems to have a simple
explanation.