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We present a numerical study on the stability of all fourth and fifth order retrograde mean motion resonances (1/3, 3/1, 1/4, 4/1, 2/3, 3/2) in the 3-body problem composed of a solar mass star, a Jupiter mass planet and an additional body with zero mass (elliptic restricted problem) or masses corresponding to either Neptune, Saturn or Jupiter (planetary problem). The fixed point families exist in all cases and are identified through libration of all resonant angles simultaneously. In addition, configurations with libration of a single resonant angle were also observed. Our results for the elliptic restricted 3-body problem are in agreement with previous studies of retrograde periodic orbits, but we also observe new families not previously reported. Our results regarding stable resonant retrograde configurations in the planetary 3-body problem could be applicable to extra-solar systems.

We present a numerical study on the stability of the 1/2, 2/1 and 1/1 retrograde mean motion resonances in the 3-body problem composed of a solar mass star, a Jupiter mass planet and an additional body with zero mass (elliptic restricted 3-body problem) or masses corresponding to either Neptune, Saturn or Jupiter (planetary 3-body problem). For each system we obtain stability maps using the n-body numerical integrator REBOUND and computing the chaos indicator mean exponential growth factor of nearby orbits (MEGNO). We show that families of periodic orbits exist in all configurations and they correspond to the libration of either a single resonant argument or all resonant arguments (fixed points). We compare the results obtained in the elliptic restricted 3-body problem with previous results in the literature and we show the differences and similarities between the phase space topology for these retrograde resonances in the circular restricted, elliptic restricted and planetary 3-body problems.

Introduction: Machine learning (ML) applications for studying asteroid resonant dynamics are a relatively new field of study. Results from several different approaches are currently available for asteroids interacting with the z2, z1, M1:2, and ν6 resonances. However, one challenge when using ML to the databases produced by these studies is that there is often a severe imbalance ratio between the number of asteroids in librating orbits and the rest of the asteroidal population. This imbalance ratio can be as high as 1:270, which can impact the performance of classical ML algorithms, that were not designed for such severe imbalances.Methods: Various techniques have been recently developed to address this problem, including cost-sensitive strategies, methods that oversample the minority class, undersample the majority one, or combinations of both. Here, we investigate the most effective approaches for improving the performance of ML algorithms for known resonant asteroidal databases.Results: Cost-sensitive methods either improved or had not affect the outcome of ML methods and should always be used, when possible. The methods that showed the best performance for the studied databases were SMOTE oversampling plus Tomek undersampling, SMOTE oversampling, and Random oversampling and undersampling.Discussion: Testing these methods first could save significant time and efforts for future studies with imbalanced asteroidal databases.