arXiv:2610.40948v1 Announce Type: cross Abstract: Rhythmic behaviours such as walking and running are commonly modelled as dynamical systems around a stable limit cycle. To keep such models tractable, it is usually assumed that each step depends only on the previous one, that the limit cycle is fixed, and that the sagittal and frontal planes can be studied separately. Here, we test these assumptions in seven public datasets of human gait (71 participants), including 16 people who both walked and ran. We fit linear step-to-step maps and validate them by cross-validation, holding out more data where more data exist. At first sight, gait has memory: the state two steps back lowers the prediction error by up to 29\%. Yet most of this temporal coupling vanishes once the operating point may drift slowly and measurement noise is modelled; gait remains limit-cycle-like, around an operating point that moves. Running keeps more memory than walking in the same people. Spatial coupling between the planes is present in most participants, mainly from sagittal to frontal, but weak. This supports planar models of healthy gait, while a repeatable, task-dependent coupling index may serve as a measure of gait quality. The simple models are wrong, but useful enough.