  The three-rotor system concerns equally massive point particles moving on a
circle subject to attractive cosine potentials of strength $g$. The quantum
theory models chains of coupled Josephson junctions. Classically, it displays
order-chaos-order behavior with increasing energy $E$ along with a seemingly
globally chaotic phase for $5.33g \lesssim E \lesssim 5.6g$. It is also known
to admit pendulum and isosceles breather families of periodic orbits at all
energies. While pendula display a doubly infinite sequence of stability
transitions accumulating at their libration to rotation threshold at $E = 4g$,
breathers undergo only one stability transition. Here, we show that these
stability transitions are associated to forward and reverse fork-like
isochronous and period-doubling bifurcations. The new family of periodic orbits
born at each of these bifurcations is found using an efficient search algorithm
starting from a transverse perturbation to the parent orbit. The graphs of
stability indices of various classes of orbits born at pendulum bifurcations
meet at $E = 4g$ forming `fans'. The transitions in the librational and
rotational phases are related by an asymptotic duality between bifurcation
energies and shapes of newly born periodic orbits. The latter are captured by
solutions to a Lam\'e equation. We also find and numerically validate values of
scaling constants for self-similarity in (a) stability indices of librational
and rotational pendula and (b) shapes of newly born orbits as $E \to 4g$.
Finally, we argue that none of the infinitely many families of periodic orbits
we have found is stable for $5.33g \lesssim E \lesssim 5.6g$, providing further
evidence for global chaos in this energy band.
