Abstract This paper discusses the asymptotic stability of the steady state and the existence of a Hopf bifurcation in discrete time multisector optimal growth models. We obtain on the one hand a local turnpike theorem which guarantees the saddle point property for all discount rates. On the other hand, we provide a new proposition which gives some conditions ensuring local stability of the steady state if the impatience rate is not too high. A characterization of the bound δ*, above which the steady state is saddle-point stable, is also proposed in terms of indirect utility function's concavity properties. On this basis, some sufficient conditions for the existence of a Hopf bifurcation are stated. We thus prove the existence of quasi-periodic optimal paths in asymmetric models.