Taming Chaos in the Three-Body Problem: Convexity Thresholds and a Bertrand–Darboux Integrability Switch

This work presents a theoretical framework for stabilizing the classical three-body problem, a famously chaotic system. We introduce a modified potential that combines Plummer regularization (to eliminate collision singularities) with harmonic confinement (to prevent particles from escaping to infinity). This modification allows the system to be transformed into a nearly integrable and predictable one under explicit, constructive conditions. Two main results are established: An explicit global convexity threshold (γ + Ω² > B(m, ε)) that guarantees stability in the sense of the KAM and Nekhoroshev theorems. This ensures quasi-periodic and predictable behavior for exponentially long times for most initial conditions. A local integrability switch based on the Bertrand-Darboux conditions. This mechanism allows the model's parameters to be fine-tuned to cancel the primary chaotic resonance near equilibrium points (such as the Lagrange points), resulting in a locally integrable system with an exact integral of motion. The framework is successfully applied to the Sun-Earth-Moon system, demonstrating that both global stability and local integrability can be achieved with physically reasonable parameters. This study offers an analytical and computationally tractable solution to "tame" the chaos in one of the fundamental problems of celestial mechanics, with potential applications in astrophysics and N-body dynamics.