How can a Poincaré map be defined for a double pendulum (or Furuta pendulum) when these systems don't have external excitations?
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Instead of a stroboscopic map, you make a Poincaré section, i.e., you define some plane (or more complex manifold) in phase space and when your trajectory intersects that plane in a certain direction, you take the intersection point as your next item of the Poincaré map.
Practically, this is often done with up- or downward zero crossings of one dynamical variable. For example for the double pendulum, you could take all moments where the main pendulum moves leftwards through its resting position. The sequence of angles and momenta of the second pendulum at those moments would be your Poincaré map (the momentum of the main pendulum is redundant due to energy conservation).
Of course, you need to take proper care to choose an appropriate plane (or similar) for your section and for some dynamics making this choice is not trivial.
