Example of a possible long-term evolution of the rotation state for (208) Lacrimosa. Rotation period P (top), osculating obliquity ε (middle-up),and longitude φ in the orbital frame associated with the s₆-frequency and s-frequency term of the nodal precession (middle-down and bottom; note φ is measured from an axis 90° away from the corresponding nodal line). The gray dots are densely output osculating values (with a time-step of 5 kyr). Black symbols in the obliquity panel are average values in a 2 Myr running window; black symbols in the bottom panels are maximum and minimum values of the respective longitude in a 2 Myr running window. The dynamical model uses solar gravitational torque and the YORP effect with parameters determined from the best-fitting solution in Sect. 2. The red lines in the upper two panels show the present state of (208) Lacrimosa for reference. At the epoch of ≃ 2.4 Gyr, the propagated spin evolution roughly matches the present state (as indicated by the gray arrows). At ≃ 1.45 Gyr (verticaldashed line 1), the solution jumps over the Slivan state of the s₆ precession frequency, where other large Koronis prograde-rotating asteroids are located. At ≃ 2.8 Gyr (vertical dashed line 2), the solution starts to closely follow the Cassini state 2 associated with the s precession frequency. This is allowed by (i) the low obliquity (where C₂ is located), and (ii) the increasing rotation period. The Cassini resonance formally bifurcates when the rotation period reaches ≃24.4 h, i.e., at ≃4.1 Gyr. Finally, at ≃5.63 Gyr (vertical dashed line 3), the small-amplitude oscillations about the resonant Cassini state 2 in the s precession frequency frame become perturbed by an overlap with the Cassini resonance associated with the s₆ precession frequency. The simulations had an initial rotation period of 12.25 h and an initial obliquity of 70°.