I assume that you are talking about an undamped undriven double pendulum.
In this case the motion exhibited by the double pendulum may be chaotic (depending on the intial conditions, lengths of the pendulum arms and masses), but it exhibits no attractors in the sense that trajectories converge to a certain invariant set (the attractor) in phase space. Instead all initial conditions are on some invariant set. For a reason, see this question of mine. So, it’s strange, but no strange attractor.
Does this pendulum, when you let it go, which can be done in many ways but let's keep the total potential energy always equal, always describe a figure that after a certain elapsed time resembles a part of a circle with a radius that's equal to the sum of the lengths of the two pendulums?
I understand the dynamics you are describing as one where both pendulum arms are always aligned and the pendulum behaves like a single pendulum. This corresponds to a regular, non-chaotic attractor. However, as already said, the double pendulum (as assumed above) does not have attractors at all. For a suited small excitation of both pendulums, you get a behaviour close to what you are describing, but it immediately starts to exhibit that motion – not just after a certain time.
To make such a dynamics happen, you would, e.g., have to have some friction within the double pendulum’s joint and nowhere else. (If you have friction elsewhere, the pendulum has a single fixed point as an attractor, namely its resting position.)