Isles of Regularity and Fractal Fate in the Gravitational Three-Body Problem: Functional Zeta-Term Analysis

The gravitational three-body problem stands as a fundamental paradigm of classical chaos, demonstrating the interplay between deterministic laws and unpredictable outcomes. Recent numerical simulations reveal multi-fractal fate maps and islands of regularity within chaotic seas. In this study, we investigate the impact of a functional zeta-term perturbation, formulated as F_zeta = G * zeta^(m) / d^2, where zeta^(m)(x) = f^m(x) - f^(m-1)(x), on the fate boundaries of the three-body problem. Our results demonstrate that, while the zeta-term can introduce minor local changes in normalized systems, its impact is negligible under physically realistic parameters. We confirm the robustness of classical chaotic fate maps and discuss mathematical, physical, and computational implications, providing a framework for future research in gravitational dynamics.The author acknowledges the use of OpenAI’s language model for assistance in drafting certain sections of this workl