>!Some reasons are common and some differ from country to country. Since other answers cover mostly common reasons, I add some reasons that are special to one or several of them. First I'd note that  I dont consider all examples mentioned below as authoritarian governments. I only say if one consider it authoritarian, it has stability because...   
  

 - `Religion:` e.g Iran. In Iran avid fans of  Ayatollah Ali Khamenei are mostly because of his religious position.  
 - `Severe repression:` e.g. [Saudi Arabia](https://www.antiwar.com/blog/2011/12/01/severe-repression-in-saudi-arabia/).
 - `In some countries there are elections that one person is the winner until dying.` e.g. Azerbaijan.  
 - `not considered as authoritarian` e.g. UK. It is widely accepted (true or false) that Queen has not notable role in the power.   
`Support of foreigners:` e.g. [Bahrain](https://en.wikipedia.org/wiki/Bahrain), One of the most sever dictatorships.  Bahrain's started large protests against its rulers in early 2011. A month later the king requested security assistance from Saudi Arabia and ... . Also Bahrain's dictator has support of [US](https://en.wikipedia.org/wiki/Naval_Support_Activity_Bahrain) and [UK](https://www.theguardian.com/politics/2016/dec/07/theresa-may-calls-on-gulf-leaders-to-press-on-with-economic-reforms).  
  
Note that countries are subject to a number of cases above; e.g. the case "Severe repression" comes true also for Bahrain.
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See also my answer to [this question](http://politics.stackexchange.com/questions/10288/why-did-arab-spring-and-iranian-green-revolution-fail-but-the-european-revoluti/10641#10641)