L1, L2, and L3 have 3 "modes" associated with them, where "mode" refers to what happens to a satellite placed near those points with just the right initial cond

L1, L2, and L3 have 3 "modes" associated with them, where "mode" refers to what happens to a satellite placed near those points with just the right initial conditions (position and velocity relative to the point). Since you can't place a satellite perfectly in practice, its motion will always be a combination of the modes, but in theory the 3 modes are: stable, unstable, and oscillatory. Like I said, since you can't place a satellite perfectly in practice, its motion will always be a combination of the 3 modes, and eventually the unstable mode will dominate and the satellite will move away from L1/2/3. BUT, if you get the math right you can position the satellite so that most of its motion is in the oscillatory mode (i.e. orbiting the point), and then as it drifts into the unstable mode you nudge it back to the oscillatory mode with a thruster firing. For context, the amount of thruster firings necessary to keep the ARTEMIS probes around Earth-Moon L1/2 was about 5m/s per year.