Ooh, a Jaime POV chapter. This should be informative!
The last time we saw Jaime he was being given booze and questioned by a grieving Catelyn who seemed to get exasperated with his answers before telling Brienne to bring her sword, which did not bode well for Jaime.
It turns out that the sword was merely used to threaten him into promising to persuade Tyrion to release Sansa and Arya before releasing him. And so Jaime finds himself on a boat with Brienne and Ser Cleos (who spends his days travelling between King's Landing and Riverrun carrying messages negotiating between Robb and Tyrion) and being chased by Ser Robin Ryger who is under orders of the Castellan of Riverrun to bring Jaime back, dead or alive, since Catelyn presumably didn't clear this scheme with the proper authorities.
There is some good banter, amidst which Jaime uses the tactic of smiling knowingly, because "men will read all sorts of things into a knowing smile, if you let them", which I must remember to find an opportunity to try.
Anyway, Brienne manages to save Jaime with some superhuman feat of scaling a cliff and hurling a boulder at Robin's boat.
And so I am reminded of an essay by John C Wright about how to write female characters. GRRM seems to be trying it all. One thing you can do is write a male character who is physically strong and good at violence and give him a woman's name: Brienne. Another approach is to realise that there is strength in femininity and have your women characters solve problems by way of diplomacy, compromise and reconciliation, which is embodied by Catelyn, who previously tried to get Renly and Stannis to shake hands and is now attempting to break a deadlock by trusting the honour of Jaime Lannister to save her daughters. I actually think there is a chance that might work, or is at least as likely to as any other possible scheme at this point.
Oh, and we now know for certain that Jaime was not behind the knife attack on Bran. He is as confused about that as anyone. It could be Cersei but Jaime thinks that would not quite be in character for her, and he would know. My current pet theory is that it is a friend of Bran's who thought he would be better off dead. Maester Luwin would be a possibility except he's now dead and such a reveal would not be dramatic enough. So which Stark could it have been?