The von Zeipel–Lidov–Kozai (ZLK) mechanism plays an important role in the long-term dynamical evolution of trans-Neptunian objects (TNOs) subjected to planetary gravitational perturbations. Despite its theoretical significance, a systematic observational census of TNOs exhibiting clear ZLK dynamics has been lacking. We performed a comprehensive search for ZLK resonances among all 1,037 numbered objects from the AstDyS catalog with semimajor axes a > 30 au. Using numerical integrations spanning up to 300 Myr, we identify 81 objects (≈7.8% of the sample) that are trapped in ZLK resonance, with an additional 23 transient objects exhibiting alternating libration and circulation. A key finding is that all TNOs in ZLK resonance are simultaneously trapped in two-body mean motion resonances (MMRs) with Neptune, providing direct empirical confirmation of theoretical predictions. The most populated MMRs are 2N-3 (58 objects), 4N-7 (15 objects), 1N-2 (9 objects), and 3N-5 (7 objects). For non-1N-2 MMRs, libration centers cluster near 90° and 270°, while the 1N-2 resonance shows shifted centers at approximately 120°, 150°, 300°, and 330°. We identify several dynamically interesting objects in ZLKR, including the retrograde TNO (585899) 2020 HM98 in 2N+9 and the distant object (652920) 2014 GR53 in the high-order 1N-18 MMR, confirming that resonances of type 1:N and high-order MMRs can maintain ZLK dynamics at large heliocentric distances.

The von Zeipel-Lidov-Kozai (ZLK) mechanism drives secular oscillations of the eccentricity and inclination that can reshape the orbital architecture of small bodies throughout the Solar System. While the mechanism has been studied in detail for trans-Neptunian objects (TNOs), giant-planet satellites, and individual near-Earth objects, no systematic census of ZLK-resonant (ZLKR) asteroids inside Jupiter's orbit has been carried out. We aim to (i) identify all numbered asteroids with semimajor axes below the orbit of Jupiter that are currently trapped in the ZLK resonance for at least 100 kyr without leaving the resonance; (ii) test whether the universal coupling between the ZLK resonance and mean-motion resonances (MMRs) found for TNOs extends to the inner Solar System; and (iii) characterize the libration centers, libration periods, and dynamical peculiarities of the ZLKR population. We integrated the orbits of all 863,471 numbered asteroids with a < 4.9 au from the AstDyS catalog using the resonances Python package and the IAS15 adaptive-step integrator in a full N-body model that includes the Sun, the eight planets, and Pluto. The resonant angle of the ZLK resonance, σ ≡ ω, was analyzed in three integration phases (100 kyr automated, 500 kyr manual, and 100 kyr MMR identification for all candidates) and classified as regular, irregular, transient, slow-circulating, chaotic, or nonresonant. Two-body MMRs up to an order of 50 and three-body MMRs up to an order of 10 were then searched for with respect to every ZLKR candidate. We identified 201 asteroids (< 0.02% of the sample) that are currently in the ZLK resonance. Of these, 163 exhibit clean libration with a stable period and amplitude (i.e., regular), 38 formally librate but display evolving parameters (i.e., irregular), and an additional 26,309 objects fall into the broader 'possible' category. The ZLKR population is strongly concentrated in the near-Earth region (144 objects with perihelion q < 1.3 au) and among Mars-crossers (28 objects with 1.3 ≤ q < 1.66 au); the main belt is almost empty. In the contrast to the TNO region, where every known ZLKR object is simultaneously trapped in an MMR, only 11 of the 201 confirmed ZLKR asteroids are in a two-body or three-body MMR; meanwhile, the MMR-ZLK coupling is not only weak but appears decoupled inside Jupiter's orbit. About one-third of the confirmed ZLKR asteroids have inclinations below the classical quadrupole critical value of ∼39°, and several confirmed librators reach c₁ = (1 − e²) cos² i > 3/5. ZLKRs within 5 au behave differently from ZLK capture in the Kuiper belt: it is an overwhelmingly planet-crossing phenomenon that is decoupled from the MMR network and it extends below the classical critical inclination. The overall number of librators is significantly lower than in the TNO region. It appears that the interplay and overlapping between different resonances in the main belt suppresses long-lived ZLK libration.

We present a systematic evaluation of modern multimodal large language models (LLMs) for the classification of mean-motion and secular resonances from images of resonant arguments. Four benchmark datasets (RB-TEST, RB-PILOT, RB-SMALL, RB-FULL) were constructed to cover clear, ambiguous, and transient cases, with both binary and three-class outputs. Using standardized prompts (a full prompt for large models and a simplified variant for small models that cannot process complex instructions), we tested flagship commercial models, large open-source models, and small locally runnable models. Commercial LLMs reach F1=100% on simple cases and up to 94% on the three-class RB-SMALL dataset, while the best open-source models also reach 100% on unambiguous cases and 76% on the complex ones. On the full binary benchmark, open-source models approach commercial performance (F1≈90–96%). Most errors occur in transient and resonance-sticking regimes. The results show that LLMs can perform resonance classification at levels comparable to those of classical or machine-learning methods without training or fine-tuning, and that even small open-source models achieve practically useful accuracy. The released benchmarks establish a reproducible standard for evaluating LLMs on dynamical astronomy tasks.

This paper presents a major enhancement to the resonances Python package that now implements full support for identifying and analyzing secular resonances. Building upon the established mean-motion resonance framework, the implementation introduces: (1) a flexible mathematical expression parser supporting arbitrary combinations of fundamental frequencies (g, s, gi, si), enabling analysis of both linear resonances (v5, v6, v16) and more than 70 nonlinear resonances from the literature; (2) specialized libration detection algorithms optimized for secular timescales, with automated parameter adaptation for extended integration times; (3) integration with existing mean-motion resonance workflows through consistent interfaces, allowing unified dynamical studies. The package has been tested through automated unit and integration tests and manual validation against examples from the literature, with all test cases---including v6, v16, z1, z2, 2v6 - v5, and 3v6 - 2v5 resonances passed successfully (with minor exceptions). The new version maintains the simplicity of the original interface, requiring only 3--4 lines of code for standard analyses, while providing researchers with powerful tools for systematic dynamical analysis and asteroid family studies. The package is available on GitHub under the MIT license.

Methods. We took orbital elements from the Minor Planet Center database and NASA JPL. For this research we used the open-source package resonances for (1) numerical integrations of orbits for asteroids over 100 000 years with planetary perturbations and (2) resonance identification. We identified all objects captured in two-body and three-body MMRs, as well as those captured in more than one resonance. Results. The study reveals that 53.76% of asteroids in the studied sample are resonant. The subset of two-body resonances contains 40.07% resonant asteroids, while the three-body subset contains 23.72% resonant asteroids. The number of asteroids trapped in two-body MMRs is significantly higher (40.07% vs. 2.0--5.0%) than previously known because of the inclusion of high-order mean motion resonances. The highest amount of two-body resonant asteroids is found at order ~36. Additionally, our results indicate that 25.57% of asteroids are involved in multiple MMRs, either through simultaneous trapping or through the phenomenon of resonance sticking, migrating from one resonance to another. Conclusions. We conclude that resonances whose order is close to the mean of all those analyzed here trap the largest number of asteroids, and that about half of the analyzed main belt asteroids are resonant. Moreover, roughly one quarter of resonant asteroids undergo migration from one MMR to another. Taken together, these results highlight the significant role of MMRs in shaping asteroid dynamics.