We consider the dynamics of a bosonic Josephson junction formed from two weakly linked atomic Bose-Einstein condensates (BECs). In particular, we study the time evolution following a sudden increase in the tunneling rate. This scenario is related to the classic problem in which two previously independent Bose-Einstein condensates are put into contact. In the two-mode approximation the model is integrable and ergodic, and the long-time probability distribution for the atom number difference between the two BECs can be calculated exactly. We examine the time that the system takes to reach this quasi-equilibrium state as well as the fluctuations around it. We also consider various extensions, such as the addition of impurity atoms which render the system chaotic. See arXiv:1202.1341 for a preprint related to this work.