What is the Ripple equivalent of the "51% attack?" In Bitcoin, if a participant gains control of more than 50% of the hash power, they can execute a double-spending attack. The "attack" could come from within, say if the existing powerful miners decide to change the rules.

Does Ripple have a similar problem, where a few people can effectively take control of the system? If yes, how does it work?
Ripple solves the double spend problem by consensus. Everyone who wants to run a server on the Ripple network picks a set of validators and tries to reach a consensus with them on which transactions are valid.

The Ripple equivalent of Bitcoin's 51% attack would be if some group obtained control over enough validators that the consensus process failed. Because people specifically select validators that they believe are unlikely to collude, this would be an exceptionally difficult thing to do.

If validators refuse to come to a consensus with each other, this is detectable to other validators. They then pronounce the network broken (because they don't know which side is right).

In this case, servers would declare the Ripple network unusable automatically. Until the problem was resolved, no transaction results could be relied upon.

The solution would consist of analyzing the signed validations and proposals to see which validators were being unreasonable and for all honest people to remove those validators from the UNLs (lists of validators they try to come to a consensus with).

So you would have to gain people's trust and then in the process of making the attack you would lose that trust and have to start over again.

By contrast, if someone acquired ASICs to attack Bitcoin, they could attack Bitcoin repeatedly. It's not clear how Bitcoin could respond to such an attack, but changing the mining algorithm (to make the attacker's ASICs worthless) is probably a non-starter as it would also make all the Bitcoin miners' investments in ASICs worthless as well.