I agree with most of what the first answer says, but I think that there is a lot more that goes into stability.
For all social castes, stability (and community and identity as well) is achieved in large part through the subconscious training that the kids are subjected to. This teaches them how to think and how to act.
For all castes, stability is also achieved through soma. People who feel that they might get to be unstable (have too much emotion) are encouraged to take soma to mellow out.
The people are also expected to achieve stability through hedonism. They are encouraged to play a lot and to have a lot of sex with a lot of partners.
Stability is mainly achieved through these methods of making sure that people do not really do much thinking.
This is especially true for the upper castes that have fewer "twins."
The world is a monolithic community. There is one World State, run by a Controller. There are no separate nations; all have been amalgamated into one. This world state controls humans much like a factory manufactures cars and a cult reaps converts: through mass production and pleasure mediation. Yes, there is an outcaste state, the Savage Reservation in New Mexico, but they don't count.
Identity is defined by the caste system. All the castes are genetically engineered according to a mold. Everyone is a clone in uniform. Mottos and mass media and conditioning are all done to teach each caste member to hate books and nature, love the state, and do his intended task without question.
Stability is achieved by the genetic engineering and birth / population control: the "Bokanovsky Process. This causes the fertilized egg to bud and create up to 96 identical individuals who will be trained to do identical jobs." This pyramid social strata creates the social hierarchy, as there is a large, mindless work force supporting the few, intelligent elite.