Advanced Techniques in Stability Analysis of Trans-Neptunian Objects

The trans-Neptunian region (30-50 AU) is a dynamically structured reservoir of icy planetesimals whose orbital architecture reflects resonant dynamics, chaotic transport, and long-term gravitational sculpting by the giant planets. This review synthesizes recent developments in the dynamical investigation of trans-Neptunian objects (TNOs), with an emphasis on mean-motion and secular resonances, as well as chaotic diffusion, in a system whose growing observational census makes it an ideal testbed for chaos detection methods. Classical indicators, including Lyapunov exponents, MEGNO, SALI/GALI, and frequency map analysis, provide the quantitative backbone for mapping TNO phase space and are complemented by modern approaches such as Lagrangian descriptors, the FAIR resonance identification method, entropy-based chaos indicators, and recurrence plot divergence methods. An anomalous diffusion framework, in which mean squared displacement scales as a power law in time, further enables classification of sub- and superdiffusive orbital transport. Machine learning has emerged as a powerful complement to traditional dynamical methods: surrogate classifiers, deep neural network solvers, and hybrid physics-data-driven frameworks together extend reliable prediction horizons in chaotic regimes and open new routes for Bayesian inference of migration scenarios. The review concludes that the most promising path forward lies in hybrid dynamical-statistical frameworks anchored to Hamiltonian dynamics, enabling efficient exploration of high-dimensional parameter spaces informed by the expanding body of trans-Neptunian observations.