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With the discoveries of more minor bodies in retrograde resonances with giant planets, such as 2015 BZ509 and 2006 RJ2, our curiosity about the Kozai-Lidov dynamics inside the retrograde resonance has been sparked. In this study, we focus on the 3D retrograde resonance problem and investigate how the resonant dynamics of a minor body impacts on its own Kozai-Lidov cycle. First we deduce the action-angle variables and canonical transformations that deal with the retrograde orbit specifically. After obtaining the dominant Hamiltonian of this problem, we then carry out the numerical averaging process in closed form to generate phase-space portraits on an e − ω space. The retrograde 1:1 resonance is particularly scrutinized in detail, and numerical results from a CRTBP model show a great agreement with our semi-analytical portraits. On this basis, we inspect two real minor bodies currently trapped in retrograde 1:1 mean motion resonance. It is shown that they have different Kozai-Lidov states, which can be used to analyse the stability of their unique resonances. In the end, we further inspect the Kozai-Lidov dynamics inside the 2:1 and 2:5 retrograde resonance, and find distinct dynamical bifurcations of equilibrium points on phase-space portraits. [ABSTRACT FROM AUTHOR]

Aims. The asteroid 2015 BZ509 is the first asteroid confirmed to be in retrograde co-orbit resonance (or 1/−1 resonance) with the giant planets in the solar system. While Saturn is the only giant planet whose trojans have not yet been discovered, we identify some small bodies among centaurs and damocloids that are potentially in 1/−1 resonance with Saturn in the present study. Methods. We integrate numerically the motion of the 1000 clones (including the nominal orbit) of each centaur whose orbit has a semi-major axis between 9.3 au and 9.8 au and an inclination i > 90°. To confirm and evaluate the 1/−1 resonant configurations mentioned above, we introduce a useful one degree integrable approximation for planar 1/−1 resonance. Results. We identify four candidates potentially in 1/−1 resonance with Saturn. The capture of candidates in this particular resonant state during the 40 000 yr integration time span is very common for 2006 RJ2 (906/1000 clones), 2006 BZ8 (878/1000 clones), and 2017 SV13 (998/1000 clones), and it is less likely for 2012 YE8 (426/1000 clones). According to our statistical results, 2006 RJ2 is the best candidate to be currently in a 1/−1 mean motion resonance with Saturn, and 2017 SV13 is another important potential candidate. Moreover, 2012 YE8 and 2006 BZ8 are also centaurs of interest but their current and long-term 1/−1 resonant state with Saturn is less likely. The proportions of the clones captured in the relative long-term stable co-orbit resonance (over 10 000 yr) are also given. The motions of the 2006 RJ2, 2015 BZ509, and 2006 BZ8 in the solar system are just around the ideal equilibrium points of the 1/−1 resonance given by the planar semi-analytical model. Conclusions. Small bodies in retrograde co-orbit resonance with giant planets may be more common than previously expected. Identification of these potential mysterious minor bodies encourages the search for such objects on a larger scale in our solar system. The findings of this paper are also useful for understanding the origin and dynamical evolution of centaurs and damocloids on retrograde orbits. [ABSTRACT FROM AUTHOR]