As you might imagine, reducing a 600 year dynasty into a quick answer is difficult. However, when discussing the stability of the empire, I have always thought that the succession methods of the Ottomans provide part of the answer to your question.

When the Sultan would have a child with one of his concubines, the mother and the child would be sent off to manage a province. This would ensure that the Sultan only had one child per concubine (so that the mother did not become a power broker in the succession lines) and that the princes would learn how to properly govern.

Once the Sultan died, the half-brothers would fight amongst each other to determine who would take over the throne. The prince had successfully killed or imprisoned his competitors would rule.

While it sounds like it would result in further turmoil and conflict, there were no potential successors left to plot a takeover once the fight was over. The winning son would prove to be the fittest and most successful leader as a result of this contest. It sounds quite gruesome, but instead of simply the eldest son ruling, the son most apt for the job won.

I'm obviously glossing over a great deal of the problems and nuances to this system of succession, so it's worth investigating on your own.

Donald Quataert, The Ottoman Empire, 1700-1922 (Cambridge: Cambridge University Press, 2005).

The wikipedia article on the subject goes into good depth as well.