Abstract Flagellar motility is critical to natural and many forms of assisted reproduction. Rhythmic beating and wave propagation by the flagellum propels sperm through fluid and enables modulation between penetrative progressive motion, activated side-to-side yaw and hyperactivated motility associated with detachment from epithelial binding. These motility changes occur in response to the properties of the surrounding fluid environment, biochemical activation state, and physiological ligands, however, a parsimonious mechanistic explanation of flagellar beat generation that can explain motility modulation is lacking. In this paper, we present the Axonemal Regulation of Curvature, Hysteretic model, a curvature control-type theory based on switching of active moment by local curvature, embedded within a geometrically nonlinear elastic model of the flagellum exhibiting planar flagellar beats, together with nonlocal viscous fluid dynamics. The biophysical system is parameterized completely by four dimensionless parameter groupings. The effect of parameter variation is explored through computational simulation, revealing beat patterns that are qualitatively representative of penetrative (straight progressive), activated (highly yawing) and hyperactivated (nonprogressive) modes. Analysis of the flagellar limit cycles and associated swimming velocity reveals a cusp catastrophe between progressive and nonprogressive modes, and hysteresis in the response to changes in critical curvature parameter. Quantitative comparison to experimental data on human sperm exhibiting typical penetrative, activated and hyperactivated beats shows a good fit to the time-average absolute curvature profile along the flagellum, providing evidence that the model is capable of providing a framework for quantitative interpretation of imaging data.