How does 'centre of mass' concept work? We know that two celestial bodies rotate around a center of mass. How does this concept work in case of solar systems where there are several planets versus one/two stars? (Even if the center of mass lies within a star, the center of mass for every planet would be different.) How is it that the rotation of celestial bodies does not become unsteady?
We know that two celestial bodies rotate around a center of mass.


"We" do not know that. Rhetorical question: Pick up an apple and drop it at high noon. Does it fall towards the solar system barycenter? The answer is of course "No." It falls away from the barycenter, towards the Earth.

I'll steal a graph from an answer I wrote to a question at the sister site, physics.stackexhange.com, Do the planets really orbit the Sun?:



Given the above graph, it appears that it's much better to say that Venus orbits the Sun (red curve) rather than the solar system barycenter (black curve). So why do people consistently use a barycentric frame rather than a heliocentric frame? The answer is simple: The barycentric frame is the frame in which the equations of motion undertake their simplest form. This is particularly so when one wants to incorporate some of general relativity into the equations of motion.

Our solar system is well-behaved. Other star systems aren't quite as well-behaved. For an example of a hypothetical system that is extremely far from well-behaved, I suggest you read Saari and Xia, "Off to infinity in finite time," Notices of the AMS.