In the last decades of the XXth century it became clear that the behavior of approximate integrals of motion called adiabatic invariants is important in studies of mean motion resonances (MMR) in dynamics of celestial bodies (e.g. [1-3]). To illustrate the opportunities of this approach, we apply it to analyze the dynamical properties of so called quasi-satellites. A quasi-satellite is an asteroid that moves in the vicinity of a planet at a distance significantly less than the distance from this planet to the Sun, and at the same time always remains outside its Hill sphere. As an example, we can mention the near-Earth asteroid 469219 Kamoʻoalewa, which is expected to be visited by the Chinese space probe Tiaiwen-2. 

Dynamically QS-mode corresponds to 1:1 MMR. We pay special attention to the secular evolution of QS-modes and to the transitions between these modes and other forms of co-orbital motions, which can occur either deterministically or (pseudo)probabilistically. 

References:
1. Neishtadt A.I.: Passage through a separatrix in a resonance problem with a slowly varying parameter, J. Appl. Math. Mech. USSR, 39, 621-632 (1975).
2. Henrard J.: Capture into resonance: an extension of the use of adiabatic invariants, Celest. Mech. Dyn. Astron., 27, 3-22 (1982).
3. Wisdom J.: A perturbative treatment of motion near the 3/1 commensurability, Icarus, 63, 272-289 (1985).