  To better understand the orbital dynamics of exoplanets around close binary
stars, i.e., circumbinary planets (CBPs), we applied techniques from dynamical
systems theory to a physically motivated set of solutions in the Circular
Restricted Three-Body Problem (CR3BP). We applied Floquet theory to
characterize the linear dynamical behavior -- static, oscillatory, or
exponential -- surrounding planar circumbinary periodic trajectories (limit
cycles). We computed prograde and retrograde limit cycles and analyzed their
geometries, stability bifurcations, and dynamical structures. Orbit and
stability calculations are exact computations in the CR3BP and reproducible
through the open-source Python package pyraa. The periodic trajectories produce
a set of non-crossing, dynamically cool circumbinary orbits conducive to
planetesimal growth. For mass ratios $\mu \in [0.01, 0.50]$ we found recurring
features in the prograde families. These features include: (1) an innermost
near-circular trajectory, inside which solutions have resonant geometries, (2)
an innermost stable trajectory ($a_{cr} \approx 1.61 - 1.85 \, a_\textrm{bin}$)
characterized by a tangent bifurcating limit cycle, and (3) a region of
dynamical instability ($a \approx 2.1 \ a_\textrm{bin}; \Delta a \approx 0.1 \
a_\textrm{bin}$), the exclusion zone, bounded by a pair of critically stable
trajectories -- bifurcating limit cycles. The exterior boundary of the
exclusion zone is consistent with prior determinations of $a_{cr}$ around a
circular binary. We validate our analytic results with N-body simulations and
apply them to the Pluto-Charon system. The absence of detected CBPs in the
inner stable region, between the prograde exclusion zone and $a_{cr}$, suggests
that the exclusion zone may inhibit the inward migration of CBPs.
