A Stability–Symmetry Approach to Periodic-Orbit Classification in the Three-Body Problem

We introduce a stability–symmetry classification scheme for periodic orbits in the planar three-body problem. The method combines three diagnostic measures, Floquet stability, discrete symmetry, and a braid-based complexity estimate, into a single quantity designed to highlight orbits with notable dynamical features. For each periodic orbit o, we compute (1) a stability index S (o) from Floquet multipliers, (2) a symmetry coefficient τ(o) reflecting time-reversal and spatial symmetries, and (3) a complexity measure C(o) derived from braid-theoretic word length. The combined indicator is defined as Q(o) = S (o) τ(o) C(o) . Applying this diagnostic to a collection of known orbits, we find that those with larger Q values tend to form a loosely defined cluster, separated from the remainder of the sample. Numerical experiments further suggest that chaotic trajectories appear to spend a considerable fraction of their evolution in neighborhoods of these elevated-Q orbits. The distribution of residence times displays approximate power-law behavior, with exponent β ≈ 9, consistent with hierarchical structure in the surrounding chaotic dynamics. These observations indicate that stability and symmetry, when considered together with topological complexity, can serve as practical tools for identifying periodic orbits that play a notable role in the phase-space dynamics. The classification scheme may provide a useful filtering step for future studies of orbit families and transport processes in the three-body problem. Keywords: three-body problem, orbit classification, stability analysis, symmetry methods, chaotic dynamics, celestial mechanics