This thesis develops new methods to analyse non-convex discrete time optimal control problems. A distinctive feature of such problems is that indifference states may occur: these are initial states at which several optimising trajectories originate. In the thesis, the genesis of such points through indifference-attractor bifurcations is studied as system parameters are varied. This necessitates an analysis of heteroclinic bifurcation scenarios of the state-costate dynamics. In particular, it is found that infinitely many indifference points exist at certain parameter values, or, equivalently, that the associated value function is not differentiable at infinitely many points in state space. The results make it possible to analyse the bifurcation structure of the discrete-time lake pollution management problem.