For GEO satellite, 24h period, it has four equilibrium points (two stable point and two unstable point). At the equilibrium point, the transverse acceleration is zero, so semimajor axis changes slowly. At stable point, satellite will oscillate around it. But at the unstable point, satellite will drift and move to the closest one of a stable point. My question is, why does satellite at the stable point just oscillate around it while at the unstable point the satellite will drift to the closest one of a stable?
Let's step away from satellites and just look at stable vs. unstable equilibrium. (I'm making use of standard examples from 1st-semester Calculus textbooks). The top of a hill is unstable equilibrium, because an object there won't move on its own. The only force is gravity, pointing straight down. But the tiniest push in any horizontal direction moves the object to a slope, and then it moves away from the top of the hill, never to return.
A stable equilibrium is a valley, or better a bowl. Again, at the very bottom, the only force is gravity, straight down. But now a small horizontal push causes the object to move uphill, and when that push is terminated, gravity pulls the object back towards the bottom of the bowl
In the absence of friction (as with the satellite), the object will continue to move up and back down in a sinusoidal motion.
But be warned - the satellite's "stable equilibrium" is a local minimum , so a large enough push will move it into an unstable non-equilibrium orbit.
As you have written, an object (infinitesimal near) near an unstable point will drift to a stable point, but it will not stop there but drift further. So an object (starting infinitesimal near) an unstable point will also oscillate around a stable point. But while the "amplitude" near a stable point is very small, an object at an unstable point has an amplitude of nearly 180 deg. it will accelerate to the stable point, "overshoot" it, be decelerated till it nearly reaches the other unstable point and go back again and again...
..::EDIT::.. (I started this as a comment but its to long and actually part of a valid answer:) @ElisaFitri the amplitude near a stable point is SMALL. So basically all objects oscillate around stable points. But for example: there is a stable point at about 108 deg. If you start at 107deg, you drift to 109 deg, and back to 107 and so on. Starting at 8 deg lets you drift to 208 deg and back. All this comes from the shape of the earth.
Looking from top earth is more elliptical than circular. So somewhere on orbit (not at the four equilibrum points at the axis) where is always "more" earth right or left from the zenit, so you will be ac-/de-celerated towards. For LEO, this compensates in one tour around earth.