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I know that the initial condition of the full double pendulum being horizontal yields motion that is not chaotic. Is there another set of initial conditions that would yield non-chaotic motion? Or do all non-horizontal initial conditions result in chaos?

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    $\begingroup$ Notice that any state reached from horizontal initial conditions can be used as non-chaotic non-horizontal initial conditions, but I know this is not the answer you are looking for $\endgroup$ – user83548 May 12 '16 at 18:08
  • $\begingroup$ Related (if not a duplicate of the second question asked therein): What is the highest energy position for a double pendulum? And for which energy is it chaotic? $\endgroup$ – Michael Seifert May 12 '16 at 18:09
  • $\begingroup$ @MichaelSeifert : Thorough as the answer in that question is, I do not think it answers this question. The answer might be in there somewhere, hidden among the technical jargon. $\endgroup$ – sammy gerbil May 12 '16 at 20:21
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Periodic orbits are possible for low energies (for which the outer limb does not flip) and very high energies (eg both limbs performing circular motion around the fixed point). In between there is a transition stage (quasi-periodic motion) towards chaos, followed by a transition back to periodic motion. I have not come across any statement of initial conditions defining the boundaries between each region, even for the 'simple' double pendulum (equal masses, equal lengths).

Examples in low-energy states :
https://www.youtube.com/watch?v=FjE-TajO-c0
https://www.youtube.com/watch?v=0ONDhqYDxx4

The following student project report gives examples of the transition from periodic to chaotic motion and back :
https://math.dartmouth.edu/archive/m53f09/public_html/proj/Roja_writeup.pdf

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