Stable Ballistic Prograde Cyclers in the Three-Body Problem
We report the first continuous families of stable, ballistic, prograde cycler orbits in the circular restricted three-body problem: periodic trajectories that alternately undergo temporary capture and orbit each primary. We construct continuous families of symmetric cyclers from intersections of the stable and unstable manifold tubes of the $L_1$ Lyapunov orbit and exhibit stable examples across more than two orders of magnitude in mass ratio, from the Sun--Jupiter regime to the equal-mass limit. Linear stability separates naturally into planar and out-of-plane components. A planar-stable branch of every computed family is created simultaneously with a hyperbolic branch in a saddle-center bifurcation of the return map at the family's maximal Jacobi constant, while out-of-plane instability occurs only through isolated parametric resonances. Every computed family contains a subfamily that is linearly stable to both planar and out-of-plane perturbations. These stable cyclers persist into the full three-body problem for sufficiently small third mass. We conjecture that saddle-center birth is universal among cycler families, suggesting that stable cyclers are a generic feature of three-body dynamics.