Glycolysis is the central pathway in energy metabolism in the majority of organisms. In a recent paper, van Heerden et al.  showed experimentally and computationally that glycolysis can exist in two states, a global steady state and a so-called imbalanced state. In the imbalanced state, intermediary metabolites accumulate at low levels of ATP and inorganic phosphate. It was shown that Baker's yeast uses a peculiar regulatory mechanism-via trehalose metabolism-to ensure that most yeast cells reach the steady state and not the imbalanced state. Results: Here we explore the apparent bistable behaviour in a core model of glycolysis that is based on a well-established detailed model, and study in great detail the bifurcation behaviour of solutions, without using any numerical information on parameter values. Conclusion: We uncover a rich suite of solutions, including so-called imbalanced states, bistability, and oscillatory behaviour. The techniques employed are generic, directly suitable for a wide class of biochemical pathways, and could lead to better analytical treatments of more detailed models. © 2014 Elsevier Inc.