We study the dynamics of the flow around a cylinder at a range of Reynolds numbers. To this end, we focus on the drag/lift phase diagram to answer the question of when the flow at the cylinder is periodic and at what critical Reynolds number the flow becomes chaotic. To this end, we consider multiple established and modern finite element approaches. These include Taylor-Hood elements with different formulations of the convective term and added stabilisation terms, exactly divergence-free and pressure-robust Scott-Vogelius elements and H(div)-conforming mass conserving mixed stress formulations. We find that most methods struggle to capture the periodic dynamics efficiently. In particular, we identify that even pressure-robust methods do not display the correct behaviour in this metric if there is no numerical dissipation in the case of under-resolved scales. Finally, we observe that the simulated drag values diverge from historic experimental values for Reynolds numbers greater than 200. We postulate that this discrepancy is either due to the three-dimensional effects or the fluid-structure interaction nature of the experimental set-up. We provide insights into ongoing work on how far these model changes help explain the observed discrepancies.